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Energy-efficient resource allocation and cooperation in wireless heterogeneous networks Ramamonjison, Rindranirina 2015

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Energy-efficient ResourceAllocation and Cooperation inWireless Heterogeneous NetworksbyRindranirina RamamonjisonB.Sc., University of Quebec at Trois-Rivieres, 2006M.Eng., Tokyo Institute of Technology, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2015c© Rindranirina Ramamonjison 2015AbstractThe deluge of mobile data demands a drastic increase of wireless network capacity. Aheterogeneous network design, in which small cells are densely deployed, is required tosatisfy this demand. However, it is critical that this dense deployment does not lead to asurge in energy cost. The aim of this thesis is to design energy-efficient resource alloca-tion methods and explore the value of cooperation in terms of energy cost. In particular,three different cooperation schemes are studied. First, a multi-cell coordination schemeis proposed for maximizing the energy efficiency of heterogeneous networks. Althoughthis problem is not convex, convergent algorithms are devised to find an efficient powerallocation. We found that this simple coordination can offer a significant energy efficiencygain even in dense networks. Second, a joint energy allocation and energy cooperationis proposed for heterogeneous networks with hybrid power sources and energy storagesystems. For this study, an offline optimization problem is considered, in which the cellsallocate their energy over time based on average rate contraints, the changing channelconditions and the fluctuating energy arrivals. It is found that an optimal use of the har-vested energy significantly improves the energy efficiency. A much larger gain is obtainedwhen energy cooperation is also leveraged, i.e. when the cells can exchange their har-vested energy through a smart-grid infrastructure. Third, the trade-off between energycost and performance is addressed for cooperative clustered small-cell networks. In thiscooperative model, the small-cell base stations form a cluster of distributed antennas tocollectively serve their mobile users. Hence, a joint optimization of cell clustering and co-operative beamforming is proposed to minimize the total energy cost while satisfying theusers’ quality of service. The problem is formulated as a mixed-integer convex programiiAbstractand solved with a decomposition method. For a given clustering, a distributed beam-forming algorithm is also designed to achieve near-optimal performance at a small cost ofsignaling overhead. Through simulations, it is shown that these algorithms converge fastand enable the cooperative small cells to save valuable energy.iiiPrefacePublications that resulted from the research presented in this thesis are as follows:• R. Ramamonjison and V. K. Bhargava, "Energy Efficiency Maximization Frameworkin Cognitive Downlink Two-Tier Networks," IEEE Transactions on Wireless Communi-cations, vol.14, no.3, pp.1468-1479, March 2015. (Linked to Chapter 2)• R. Ramamonjison and V. K. Bhargava, "Sum energy-efficiency maximization for cog-nitive uplink networks with imperfect CSI," Wireless Communications and NetworkingConference (WCNC), 2014 IEEE, pp.1012-1017, April 2014. (Linked to Chapter 2)• R. Ramamonjison and V. K. Bhargava, "Energy Allocation and Cooperation for Energy-efficient Wireless Two-Tier Networks," Submitted to a journal for publication consid-eration, 2015. (Linked to Chapter 3)• R. Ramamonjison, A. Haghnegahdar and V. K. Bhargava, "Joint Optimization of Clus-tering and Cooperative Beamforming in Green Cognitive Wireless Networks," IEEETransactions on Wireless Communications, vol.13, no.2, pp.982-997, February 2014.(Linked to Chapter 4)• R. Ramamonjison and V. K. Bhargava, "Distributed beamforming in cognitive multi-cell wireless systems by fast interference coordination," Personal Indoor and Mo-bile Radio Communications (PIMRC), 2012 IEEE 23rd International Symposium on,pp.2208-2213, Sept. 2012. (Linked to Chapter 4)I am the primary author of these publications and has had the main responsibility in devel-oping the original ideas, the mathematical analysis and the Matlab programs used for theivPrefacecomputer simulations. The contributions of my co-authors are as follows. Prof. Vijay K.Bhargava is my honorable supervisor and he provided valuable guidance and directions inidentifying the research problems as well as constructive feedback during the preparationof the manuscripts. Alireza Haghnegahdar was a former Master student in our researchgroup; he contributed in one of the above publications by offering important technicalfeedback during the formulation of the research problem. He also helped in running someof the simulations and offered editorial support when preparing the manuscript for publi-cation.Some of the simulation results involving constrained optimisation were obtained usingthe disciplined convex optimization software CVX developed by Grant, Boyd & Ye [1].vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Energy Efficiency Optimization in Wireless Two-tier Networks . . . . . . . . . 122.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15viTable of Contents2.4 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Special Case: Orthogonal Multi-cell Transmissions . . . . . . . . . . . . . . 212.6 General Case: Non-orthogonal Transmissions . . . . . . . . . . . . . . . . . 272.7 Power Allocation with Probabilistic Interference Constraints . . . . . . . . . 332.8 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Energy Allocation and Cooperation for Hybrid-Powered Two-tier Networks . 473.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.4 Energy Allocation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.5 Joint Energy Allocation and Energy Cooperation . . . . . . . . . . . . . . . 643.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704 Power minimization of Cooperative Clustered Small-cell Networks . . . . . . 774.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.4 Joint Clustering and Beamforming Optimization . . . . . . . . . . . . . . . 834.5 Distributed Multi-cell Cooperative Beamforming . . . . . . . . . . . . . . . 954.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045 Summary, Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . 1105.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131viiTable of ContentsA Proof of Results in Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132A.1 Proof of Proposition 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132A.2 Proof of Lemma 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134A.3 Proof of Proposition 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137B Proof of Result in Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140C Proof of Results in Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142C.1 Proof of Lemma 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142C.2 Proof of Proposition 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143C.3 Proof of Lemma 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144C.4 Proof of Lemma 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144viiiList of Tables2.1 Simulation parameters in Chapter 2. . . . . . . . . . . . . . . . . . . . . . . 373.1 List of key parameters in Chapter 3. . . . . . . . . . . . . . . . . . . . . . . . 503.2 Path loss model and system parameters in Chapter 3. . . . . . . . . . . . . . 714.1 List of key parameters in Chapter 4. . . . . . . . . . . . . . . . . . . . . . . . 824.2 Overhead per iteration of limited signaling schemes. . . . . . . . . . . . . . 1034.3 Power cost and system parameters in Chapter 4. . . . . . . . . . . . . . . . . 104ixList of Figures1.1 Centralized (left) and distributed (right) heterogeneous RAN architectures. . 61.2 Rate and energy efficiency vs. power (dB) . . . . . . . . . . . . . . . . . . . 72.1 Wireless two-tier network model. . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Cell association regions of a two-tier network with 5 macro-cell users and10 small-cell BSs. Solid and dashed circular lines show the cell edges witheffective cell association bias β of 6 dB and 9 dB respectively. Small-cellusers are randomly located on the cell edges. . . . . . . . . . . . . . . . . . 392.3 Convergence of proposed algorithms. . . . . . . . . . . . . . . . . . . . . . . 402.4 Average sum energy efficiency comparison for different number of smallcells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.5 Average sum rate comparison for different number of small cells. . . . . . . 422.6 Average total interference power at a macro-cell user. . . . . . . . . . . . . . 432.7 Performance of sum rate-optimal power allocation vs. Imax with P0 = 10 dB. 442.8 Performance of energy-efficient optimal power allocation vs. Imax for differ-ent P0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.1 Model for renewable energy management. . . . . . . . . . . . . . . . . . . . 523.2 A model of cellular networks with hybrid power source and energy sharingthrough a smart micro-grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.3 Base stations and user locations. . . . . . . . . . . . . . . . . . . . . . . . . 70xList of Figures3.4 Channel gains for the desired signal (solid line) and inter-cell interference(dotted line) received by SUs. . . . . . . . . . . . . . . . . . . . . . . . . . . 723.5 Convergence of Algorithm 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . 733.6 Comparison of sum energy efficiency for different energy harvesting ratesand arrivals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.7 Energy efficiency per cell for energy-harvesting rates between 0 dB and 10 dB. 753.8 Grid power consumed per cell for different Poisson arrival rates. . . . . . . . 764.1 Coordinated multi-point transmission model in distributed and cognitivesmall-cell networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2 Generalized Benders Decomposition procedure for joint BS assignment andmulti-cell beamforming problem. . . . . . . . . . . . . . . . . . . . . . . . . 924.3 Optimal clustering for different network topologies and SINR targets . . . . 1054.4 Performance comparison and effect of SINR requirements of small-cell users. 1064.5 Effect of the number of small-cell users with 10dB SINR target . . . . . . . 1064.6 Convergence of proposed algorithms . . . . . . . . . . . . . . . . . . . . . . 1074.7 Cooperative network with 3 clusters and |U| = 14, |V| = 3. . . . . . . . . . . 1084.8 Performance comparison of limited signaling schemes with I = 3dB. . . . . 108xiList of Algorithms2.1 Algorithmic procedure for solving (P ) . . . . . . . . . . . . . . . . . . . . . . 272.2 Minorization-Maximization algorithm based on Newton method . . . . . . . 303.1 Offline energy allocation algorithm for solving (P ) . . . . . . . . . . . . . . . 583.2 Solving the feasibility problem (F ) . . . . . . . . . . . . . . . . . . . . . . . . 603.3 Subprocedure for solving the surrogate problem (Qq). . . . . . . . . . . . . . 643.4 Algorithm for solving the energy management and cooperation problem (C) . 69xiiList of AbbreviationsADMM Alternating Direction Method of MultipliersAWGN Additive White Gaussian NoiseBS Base StationC-RAN Centralized Radio Access NetworkCCCP Convex-Concave ProcedureCoMP Coordinated Multi-PointCSI Channel State InformationCU Central UnitDC Direct CurrenteICIC Enhance Inter-Cell Interference CoordinationGBD Generalized Benders DecompositionMBS Macrocell Base StationMIMO Multiple-input multiple-outputMM Minorization-MaximizationMU Macrocell UserRAN Radio Access NetworkxiiiList of AbbreviationsRRU Remote Radio UnitSBS Small-cell Base StationSC Small CellSINR Signal-to-Noise Power RatioSU Small-cell UserxivAcknowledgementsI feel extremely fortunate to have met and learned from many incredible people at theUniversity of British Columbia during my Ph.D. study. Foremost, I would like to thankmy advisor Professor Vijay K. Bhargava for his guidance, patience and support over thepast four years. I am very grateful for the tremendous opportunity he has given me.Professor Bhargava’s exceptional motivation, vision, generosity and integrity will alwaysbe an inspiring role model for my future career.I would also like to thank Professor David Michelson, Professor Vincent Wong andProfessor Alireza Nojeh for serving as committee members during my PhD qualifying ex-amination and for offering practical and constructive feedback.My gratitude also goes to my colleagues at the CITR research group, who offered metheir invaluable support, collaboration, and friendship. Special thanks go to Alireza Hagh-negahdar with whom I had the good fortune to collaborate. I have greatly benefited fromthe many hours of stimulating discussions I had with Alireza Haghnegahdar, Roya ArabLoodarichech and K. N. R. Surya Vara Prasad. I also appreciate very much the enor-mous advice and help that I received from my senior colleagues namely Umesh Phuyal,Zia Hasan, Shankhanaad Mallick, Hamidreza Boostanimehr and Rajiv Devarajan. I amdeeply indebted to my colleagues who boldly reviewed this dissertation and my papers (inchronological order): Alireza Haghnegahdar, Shristi Pradhan, Roya Arab Loodarichech, K.N. R. Surya Vara Prasad, Sudha Lohani and Buddhika Nettasinghe. I also thank our guestvisitors Frederik Klement and Dr. Ning Wang for the engaging discussions I had with them.Special thanks go to the faculty and staff at the UBC ECE department for their greatsupport, including Professor Lutz Lampe, Sonia Dhillon, Brandie Hill, Kristie Henriksen,xvAcknowledgementsDavid Chu Chong, Sabrina Ho and many others.I wish to thank also my previous mentors whom I had the chance to meet before andduring my graduate studies. I would like to thank Professor Adel Omar Dahmane, Dr.Patrick Marsch and Professor Gerhard Fettweis for introducing me to the fascinating worldof wireless communications. I also wish to express my gratitude to Professor KiyomichiAraki and Professor Kei Sakaguchi who welcomed and supervised me during my Masterstudy and encouraged me to pursue a Ph.D. study.Last but not the least, I feel very blessed to have my dear parents, my beloved wife andmy extended family who not only offered me their love and support but also sacrificeda lot when I have been studying abroad. I would like to thank them for being patient,encouraging and caring during all these years.This Ph.D. thesis would not have been possible if not for the genereous financial sup-port that I received from the Four Year Fellowship program and from Professor Vijay K.Bhargava. This work was supported in part by grants from the Natural Sciences and Engi-neering Research Council of Canada (NSERC).xviTo my parents, my wife and my son,with gratitude for your inspiration, love and support.xviiChapter 1IntroductionThe efficient allocation of resources is a constant and primary concern in the design andoperation of wireless cellular networks. Reliable wireless communications fundamentallydepend on two key resources: radio frequency spectrum and energy. Since the early daysof mobile telephony, it is recognized that the spectrum is a very scarce resource and mustbe efficiently reused [2]. In response, the cellular concept was invented at Bell Labs in1947 as an effective way to improve the spectrum efficiency, expand the service coverageand increase the network capacity. This is done by installing more base stations, splittingexisting cells into smaller ones [3] and by reusing the spectrum across distant cells. Infact, cell splitting is still one of the main drivers for capacity growth in all four generationsof cellular networks [4].Today, network operators face a greater challenge since the proliferation of smart-phones and the explosion of mobile multimedia applications have led to an exponentialgrowth of mobile data traffic. Unfortunately, this trend is expected to continue as emergingInternet-of-Things applications will further amplify this traffic growth. Such applicationswill drive more physical objects, such as transportation vehicles, health monitoring systemsor even construction diggers, to be remotely connected to the Internet [5]. In the light ofthese data-hungry applications, the wireless industry faces a tremendous challenge to sup-port a 1000-fold increase in traffic demand over the next decade [6].Such capacity crunch will require an intense densification of the networks [4]. Thistrend will result in a heterogeneous multi-tier design, in which a mixture of small cells andmacrocells is used. The small cells are deployed over macrocells to upgrade the capacity in1Chapter 1. Introductionurban and densely populated areas. As a result, each user is brought closer to the network,ideally within 10-30 meters of the closest small-cell base station. This heterogeneousdesign has many benefits, which include lowered link power budget, decreased load percell, higher link data rate and better indoor coverage [7].However, the dense deployment of small cells also raise important questions aboutthe energy efficiency of such networks. Even if a small cell is served by low-power basestations, the high density of base stations may lead to a surge in energy cost. Recently,operators have realized the importance of managing their networks in an energy efficientway. In fact, studies show that information and communications technology infrastructurecontributes to a 2% of the total amount of CO2 emission levels [8]. Given the exponentialincrease in data traffic, it becomes critical to maintain the energy cost at its current level.This is possible only if more emphasis is placed on energy efficiency when designing andoperating future networks.In spite of these pressing needs, there is a lack of understanding on how to improvethe energy efficiency of cellular networks. For example, the implication of heterogeneousnetworks on energy cost is not well understood. In fact, previous works have mainlyfocused on the rate maximization problem. Although there exist some studies on energy-efficient resource allocation, they have considered only the simplest cases such as a point-to-point system [9] or a single-cell downlink system with orthogonal transmissions [10,11]. In contrast, maximizing the energy efficiency in a multi-cell multi-user scenario, suchas small-cell networks, is a challenging open problem. This problem is further complicatedby the inevitable presence of the interference due to the need for full spectrum reuse. Thus,new methods are needed to maximize the energy efficiency of heterogeneous networks.In addition, network operators also need new sustainable energy solutions to increasethe capacity in a cost-effective manner. An attractive solution is to harvest inexpensiveand clean energy at each cell from renewable sources such as solar, wind or even radio-frequency waves [8]. Given the fluctuation of the harvested energy, it is required to com-2Chapter 1. Introductionbine the renewable sources with a conventional power source to avoid a detriment of thequality of service. Actually, this hybrid approach has already been applied to power net-works in remote or rural areas [12]. For example, network operators have relied on solarpanels, batteries and diesel generators due to lack of consistent access to electricity [12].Nonetheless, further research efforts are needed to study the efficient use of hybrid sourcein large-scale heterogeneous networks. Previous works on resource allocation with hybridsources have only considered very simple scenarios [13–15].Another important consideration is the use of cooperation schemes among the cells.In fact, there are three ways in which cells can cooperate. The first cooperation schemeis coordinated multi-cell power allocation. The second cooperation is multi-cell energysharing, in which the cells can exchange their harvested energy. This form of cooperationis motivated by recent efforts by the power industry to modernize electrical power sys-tems. In particular, future smart-grid technologies will enable a bidirectional energy flowbetween distributed energy generators and consumers [16]. This new capability shouldbe leveraged to better utilize the limited energy in future heterogeneous networks. Thethird cooperation scheme is based on coordinated multi-point (CoMP) transmission, inwhich the cells share their antennas to collectively serve their users. Using this cooper-ative transmission scheme, the small cells can better mitigate the interference, achieve aspatial multiplexing gain and counter the negative effects of the wireless channel fading.Although many works have explored the benefits of CoMP for spectrum efficiency improve-ment [17–23], its value in terms of energy efficiency is not yet completely understood.This thesis deals with the design of energy-efficient resource allocation methods usinga mathematical programming approach. Given that base stations are the most power-hungry equipment in cellular networks [8], we focus on the downlink of heterogeneousnetworks with multiple cells and multiple users. Our goal is to answer the followingtwo questions. How to achieve energy-efficient transmissions in a multi-cell multi-userenvironment? What is the value of cooperation when the cells share their energy or their3Chapter 1. Introductionantennas?To do that, we consider three specific network models, each mainly characterized bythe kind of cooperation used by the cells. These models are chosen in order to exploredifferent solutions for enabling energy-efficient heterogeneous networks, and to capture avariety of deployment and cooperation scenarios. The first model is a small-cell networkwith a centralized architecture. Specifically, the cells are served by radio remote units,which are connected to a central unit through fronthaul fiber links. In this scenario, thepower allocation of the cells is coordinated by the central unit to maximize the sum oftheir energy efficiencies. The second model is a heterogeneous small-cell network with ahybrid power source. Precisely, each cell is powered by both a conventional power gridand by renewable sources. In addition to this hybrid power supply, an energy cooperationis also used so that the cells can exchange their harvested energy through a smart-gridinfrastructure. In this second scenario, the cells also maximize their energy efficiency byefficiently allocating the pool of harvested energy over a finite time horizon. Finally, thethird model consists of distributed small-cells with conventional power source. The cellsdo not share their energy but they can cooperate by sharing their antennas and jointlytransmitting to their users. In contrast to energy cooperation, this type of cooperation isused to improve the performance at the expense of energy consumption. In this case, thecells should minimize the total energy cost by optimizing the degree of cooperation andtheir beamforming design given a target quality of service.In the rest of this chapter, the model of heterogeneous networks is described, the con-sidered resource allocation and multi-cell cooperation schemes are presented in more de-tails, and the main results of the thesis are outlined.41.1. Background1.1 Background1.1.1 Wireless Heterogeneous NetworksA heterogeneous multi-tier design will play an important role to support the escalatingtraffic demand of future mobile applications [6]. In heterogeneous multi-tier networks, alarge number of small cells are added to complement the macrocells. These small cells canbe of different sizes and include microcells, picocells, indoor femtocells as well as remoteradio units.There are several benefits to adding small cells [24]. First, they offer a capacity boost inhot spots with high traffic demand. Next, they improve the coverage in areas not coveredby the macrocells both outdoors and indoors. They also improve the overall networkperformance by offloading the traffic from the large macrocells. As a result, heterogeneousnetworks can offer a much higher peak and average rates per user as well as a higher bitrates per unit area.Unfortunately, a heterogeneous network architecture also comes with many technicalchallenges [24, 25]. As explained earlier, it is crucial to minimize the energy cost of thenetwork in spite of the dense small-cell deployment. Because the spectrum is also limited,it must be fully reused while mitigating the resulting interference between the cells to anacceptable level. Another drawback of a heterogeneous design is the need to connect thesmall cells’ base stations to the core network through backhaul or fronthaul links [24].Given different usage and deployment scenarios, both centralized and distributed ar-chitectures have been envisioned for heterogeneous networks [24]. These are illustratedin Figure 1.1.Distributed network architectureIn a distributed deployment scenario, each small cell has a standalone access point or basestation. Each base station has its own power supply. The radio access network functions51.1. BackgroundMacro Base StationRemote Radio UnitCentral Unit Fronthaul optical linkBackhaul optical or mmWave linkSmall-cell Base StationFigure 1.1 – Centralized (left) and distributed (right) heterogeneous RAN architectures.of each cell, such as user scheduling, resource allocation and baseband processing, areperformed by its own base station. In this case, backhaul links are needed to connect eachcell to the core network. When coordination is needed between the cells, some signalinginformation must also be exchanged through backhaul links. This backhaul infrastructurecan be implemented either with optical fiber links or with wireless millimeter-wave linkswith a large bandwidth [26].Centralized network architectureRecently, there is a strong interest to deploy small cells using a centralized radio accessnetwork architecture (C-RAN), also known as “cloud” RAN. In this case, each cell is servedby a remote radio unit (RRU). The remote units are then connected to a central unit(CU) via fronthaul optical fiber links. In contrast to the distributed deployment, the radioaccess network functions of the cells are performed by the central unit. As a result, theenergy consumption due to processing is controlled by the central unit. This centralizedarchitecture facilitates the coordination between the cells and can also improve the energyefficiency of the network [27, 28].61.1. Backgroundlog(P)  Data RateEnergy EciencyFigure 1.2 – Rate and energy efficiency vs. power (dB)1.1.2 Resource Allocation Problems in Wireless NetworksThe efficient usage of the spectrum and energy is critical to improving the overall expe-rience of mobile users and to reducing the operational costs of the network. Typically, aresource allocation problem is first formulated as an optimization problem based on spe-cific objectives and constraints. Then, algorithms must be developed to find a feasible andefficient solution to the optimization problem.There are several possible design objectives in wireless communications. To give asimple illustration, let us first consider a single-user single-carrier communication systemwith a bandwidth W , a transmit power P and a signal-to-noise power ratio gσ2 .A common design goal is to maximize the Shannon’s channel capacity C given by [29]:C = W log2(1 +gPσ2). (1.1)This capacity function, which defines the maximum achievable data rate for reliable com-munications is a strictly increasing function of the transmit power.71.1. BackgroundFrom energy-limited systems, it is instead desirable to either minimize the energy con-sumption necessary to satisfy a given rate requirement or to maximize the energy effi-ciency, also known as the bits-per-Joule capacity [9, 30]:EE =WP0 + ψPlog2(1 +gPσ2). (1.2)In (1.2), P0 is the processing power dissipated in the circuit blocks and ψ is the poweramplifier efficiency. In contrast to the capacity (1.1), the energy efficiency function (1.2)is not a monotonically increasing function of the transmit power. As shown in Figure 1.2,there is a point at which the effective amount of data transmitted per unit of power is max-imum. Another important difference between these two functions is that the rate utilityfunction is concave whereas the energy efficiency function is not. More precisely, (1.2)is a quasi-concave function. This subtle difference makes the energy efficiency maximiza-tion problem much more difficult to solve, especially in a multi-user interference-limitedscenario. In fact, it has been solved only in simplest scenarios, such as single-user system[9] or in single-cell downlink system with orthogonal multi-access scheme [10]. Unfor-tunately, the algorithm used [9] and [10] cannot be applied to a multi-cell environment,such as heterogeneous networks.As previously mentioned, this thesis focuses on the downlink of cellular networks be-cause the energy consumption of base stations is of a greater concern to network operators.Note that our design objective is different from that of wireless sensor networks. Sincesensors are powered solely by batteries, the main concerns in these systems are rather tominimize the transmission delay or to maximize the expected amount of data transmitteduntil the sensor’s battery dies [31].81.1. Background1.1.3 Coordination and Cooperation Schemes in HeterogeneousNetworksIn a heterogeneous network, the coordination and cooperation between the cells has apositive impact on the performance in multi-cell environments. We consider three types ofcooperation in this thesis.First, multi-cell coordination is beneficial for power control and interference manage-ment. In fact, significant research efforts have been devoted to develop enhanced inter-cellinterference management (eICIC) schemes in heterogeneous networks [32–35]. This is dueto the fact that traditional frequency reuse and orthogonal multi-access schemes are inef-ficient or infeasible for a large-scale and heterogeneous deployment [32]. To facilitate thecoordination between cells, a centralized network architecture, such as a C-RAN based de-ployment is desirable. In fact, a distributed coordination is more complicated as signalinginformation needs to be exchanged between the cells. In this case, the signaling overheadshould be kept at a minimum level due to the limited capacity of the backhaul links [36].Second, coordinated multi-point transmission or CoMP is a more powerful cooperationscheme in which the small-cell base stations form a cluster in order to jointly transmit thedata to each user [37]. It can be used both in downlink and uplink. When CoMP is used ina heterogeneous network, a number of cells is involved in the data transmission to or fromeach user. However, CoMP requires that the transmissions are synchronized and that theuser data are available at every point. Hence, it requires backhaul or fronthaul links witha higher capacity. Moreover, a C-RAN based deployment can facilitate its implementation[38]. However, this form of cooperation can incur a large energy consumption due to themulti-point processing.Third, an energy cooperation scheme can also be envisaged for distributed small cellsthat are powered by a hybrid power source, i.e. each cell draws energy from both re-newable and conventional energy sources. However, previous works that considered thedynamic energy management of hybrid-powered cellular networks only considered the91.2. Outline of Thesissingle-cell scenario [10, 39, 40]. For example, [10] studied the energy-efficient power al-location for a single-cell downlink system. [40] proposed a detailed model of smart hybridpower system and a real-time energy management to reduce the operational cost of abase station. Given the recent developments of smart-grid technologies [16, 41], it willalso be important to explore the future possibility for the cells to exchange their harvestedrenewable through the smart-grid [42].1.2 Outline of ThesisThe focus of this thesis is to explore the design of energy-efficient resource allocationmethods as well as the value of cooperation in heterogeneous multi-cell networks. Theoutline and the contributions of each chapter are as follows.Chapter 1, this present chapter, gives the motivation, background and contributions ofthis thesis.Chapter 2 develops a simple multi-cell coordination scheme for maximizing the energyefficiency of wireless heterogeneous networks. Precisely, each cell optimizes their powerallocation to transmit the maximum number of bits per unit of energy. In this optimizationproblem, a wide variety of power constraints can be accommodated including determinis-tic and probabilistic interference constraints. In addition, two scenarios with orthogonaltransmissions and full spectrum sharing are both treated. Using a parametric approxima-tion approach, the original complicated non-convex problem is related to a much simplerfamily of convex parametric problems. Given this result, the energy-efficient power allo-cation problem can then be solved within the powerful framework of convex optimizationtheory. In particular, convergent algorithms are derived and proven to converge to globaloptimum for the case of orthogonal transmissions, and to at least a local optimum for thecase of full spectrum sharing. These algorithms are tested in a simulation of small-cell net-101.2. Outline of Thesisworks. The simulation results show that the proposed algorithms have a fast convergenceand achieve much better performance compared to an existing game-theoretical scheme.Chapter 3 considers the joint energy allocation and energy cooperation for heteroge-neous networks with hybrid power sources. We assume that each cell has access to arenewable energy source and to the conventional power grid; and has also an energystorage system. The cells thus allocate their energy over time by considering the time-varying channel conditions and energy arrivals. The objective is to maximize the energyefficiency subject to average rate constraints, battery limit constraints and energy causalityconstraints. Furthermore, an energy cooperation scheme, in which the cells can exchangetheir harvested renewable energy through a smart-grid infrastructure, is also considered.The goal of this study is to investigate the benefits of using hybrid power source and energycooperation scheme in heterogeneous networks. Therefore, offline energy-efficient powerallocation algorithms are developed for this goal. To do that, the optimization frameworkof Chapter 2 is extended to handle the dynamic energy constraints and the non-convexrate constraints.Chapter 4 considers the joint clustering and cooperative beamforming optimization inheterogeneous network. In this model, the cells cooperate not by sharing their energy butinstead by sharing their antennas. In other words, they perform a coordinated multi-pointtransmission by forming a cluster of distributed antennas. This multi-cell cooperation cangreatly improve the performance of the cells at the expense of an increased energy cost.Thus, the focus of this chapter is to develop practical methods to achieve an efficient trade-off between energy and performance. Specifically, a convergent algorithm is proposed forfinding the joint optimal clustering and beamforming vectors. In addition, a distributedmulti-cell beamforming with limited signaling is designed for a given clustering. The con-vergence and performance of these algorithms are analyzed through simulations.Finally, Chapter 5 summarizes the main results of the thesis and indicate some inter-esting directions for future research.11Chapter 2Energy Efficiency Optimization inWireless Two-tier Networks2.1 IntroductionIn this chapter, we present a generic optimization framework for the energy-efficient powerallocation in wireless heterogeneous two-tier networks. The two-tier base stations controltheir transmission powers to maximize the sum of their energy efficiency while respectingvarious power constraints. Moreover, we assume that the two-tier network can share thespectrum with a primary system.The main contribution in this chapter is an optimization framework for maximizingthe sum energy efficiency of the two-tier cells subject to shared and individual power con-straints. Before presenting the system model in Section 2.3, we give a detailed review ofstate-of-the-art research in energy efficient resource allocation in Section 2.2. In contrastto earlier works, we focus on the more difficult multi-cell energy-efficient power allocationproblem. We solve this non-convex problem by looking at both the orthogonal multi-access and full spectrum reuse scenarios. In Section 2.5, we first assume that the cells’transmissions are non-interfering or orthogonal. Despite the non-convexity of the prob-lem, we show that it is possible to exploit the problem structure using convex parametricprogramming. Then, we derive an algorithm which converges to the global optimal powerallocation.122.2. Related WorksNext, we relax the orthogonality constraint in Section 2.6 and allow the cells to inter-fere with each other while sharing the spectrum. Another algorithm that monotonicallyconverges to at least a local optimum is derived by applying the minorization-maximizationprinciple in conjunction with the Newton method.We also extend the application of the optimization framework to energy-efficient powerallocation problems with imperfect channel state information between the two-tier basestations and the primary users. In this case, the interference power constraints becomeprobabilistic. In Section 2.7, we show how to handle such probabilistic constraints usingdifferent robust approximation methods.Finally, we present some simulation results in Section 2.8, which validate the conver-gence of these algorithms. In addition, we compare their performance against existingschemes and analyze the effect of network parameters on the energy efficiency.2.2 Related WorksThe design of energy-efficient resource allocation schemes is not a trivial task mainly be-cause of the non-concavity of the energy efficiency utility function.As mentioned in Chapter 1, the energy efficiency maximization was studied only undersimplified network scenarios. First, a single-user point-to-point setting was considered by[9] and [43]. In [9], Isheden et. al. showed that the energy efficiency function is pseudo-concave for a single-user system. As a result, the simple Dinkelbach algorithm [44] canbe used to find the optimal solution. The Dinkelbach method could be further applied tomore complicated scenarios under two conditions. First, the utility function must consistof a single ratio of the sum rate over total power consumption. Second, the multi-usertransmissions should happened over orthogonal non-overlapping channels. For instance,this approach was used in [45] and [46] for the orthogonal single-cell downlink systemand in [47] for the single-cell uplink system. Similarly, [48] used these two assumptions132.2. Related Worksand considered different utility functions for the downlink and uplink systems. For theuplink system, the considered design objective is to maximize the minimum energy effi-ciency among all the users. In addition, [48] proposed low-complexity algorithms to findsuboptimal solutions. In [49], the downlink energy optimization of a simultaneous wire-less information and power transfer system is studied. Again, [49] assumed orthogonaltransmissions. This study was then extended by [50], which also takes the imperfectionsof the channel estimation and the multi-user scheduling into account. Finally, similarenergy-efficient resource allocation schemes were proposed for OFDMA relay networks.For example, [51] solved a joint relay selection, subcarrier allocation and power alloca-tion problem. The Dinkelbach method was leveraged with an integer relaxation and dualdecomposition to derive practical algorithms. Furthermore, the cooperative beamformingproblem was studied in [52] for a virtual MIMO system consisting of a single source node,a single destination node and several distributed relay nodes. [52] analyzed the numberof relay nodes that maximizes the efficiency for a given outage constraint. In addition, amode-switching algorithm between cooperative beamforming and direct transmission wasstudied. Finally, energy-efficient beamforming and scheduling schemes were proposed in[53] for a coordinated multi-cell downlink network. In that model, a zero-forcing beam-forming method was used to enforce orthogonality between the transmissions. [53] alsochose the single ratio of sum rate over sum power consumption as the objective function.These aforementioned works share two basic assumptions: orthogonality betweentransmissions is enforced and the energy efficiency utility metric consists of a single frac-tion. While such a utility function is perfectly valid for a single-cell downlink transmission,it is not applicable for multi-cell multi-user systems such as a heterogeneous two-tier net-work. Instead, a more practical utility function is the sum energy efficiency of all users.In fact, the base station of each cell has independent power amplifier and energy source.In such setting, the cells also compete for resources. Therefore, it is more natural to usethe sum energy efficiency metric. Actually, it was also shown in [54] that the sum energy142.3. System Modelefficiency utility provides higher degrees of freedom for the system design than the ratioof sum rate and total power consumption.However, maximizing the sum energy efficiency is a very hard problem since this util-ity function is not even quasiconcave. Hence, the Dinkelbach algorithm can no longerbe applied. In addition, the limited spectrum must be reused in heterogeneous two-tiernetworks. In fact, it was shown in [55] that the resource allocation becomes inefficientwhen channel orthogonality is enforced in two-tier networks. As a result, the interferencebetween the cells must be mitigated. Nonetheless, the multi-user interference also com-plicates the resource allocation problem. The previous work in [56] circumvented thisdifficulty by aiming for a Nash equilibrium. In contrast, we propose in this chapter a noveloptimization framework that maximizes the energy efficiency maximization in multi-cellmulti-user environments. The framework can handle non-orthogonal transmissions. Incontrast to [56], it achieves at least a local optimum.2.3 System ModelIn this chapter, we consider the downlink of a wireless two-tier network shown in Fig.2.1. Thus, the system is composed of the two-tier cells C ={c1, . . . , c|C|}and of the usersU ={u1, . . . , u|U|}. Here, the notation |S| denotes the cardinal of a set S. Each base station(BS) of a cell c serves a subset of users Uc. We assume a centralized radio access networkarchitecture (C-RAN), in which each small cell is served by a radio remote unit (RRU). TheRRUs are connected to a central unit (CU) through fronthaul optical or millimeter-wavelinks. The resource allocation is centralized at the CU. In the rest of this chapter, we usethe terms RRU and small-cell base station (SBS) interchangeably. We assume that the BSsand the users are equipped with single antennas. Moreover, the multi-cell transmissionsare performed over a set of subcarriers S = {1, . . . , N}.152.3. System Model   Small-cell User (SU) Primary User (PU)Macro Base Station (MBS) Remote Radio UnitCentral Unit Fronthaul link Desired signal InterferenceFigure 2.1 – Wireless two-tier network model.The power allocation vector of all BSs is denoted by the vector p = (pu,n)u∈U , n∈S ∈RN |U|+ . Thus, the signal-to-interference-plus-noise ratio (SINR) of user u on subcarrier n isgiven by:SINRu,n (p) =pu,nguu,n∑j∈U\{u}pj,nguj,n + σ2u,n, (2.1)where σ2u,n is the variance of an additive white Gaussian noise (AWGN) at user u. In(2.1), guj,n denotes the channel gain between user u and the BS serving another user j.It accounts for the path loss attenuation, the large-scale shadowing and the small-scaleRayleigh fading with distribution CN (0, 1) between the BS and the user.Remark 2.1. We assumed a Rayleigh distribution to model the fading envelope distributionof both the desired and interference channels. This is a somewhat conservative assumptiongiven that, in some small-cell environments such as indoor femto-cells, there can be adominant line-of-sight path between the BS and the user. Nonetheless, our assumptionis meant to simplify the analysis. In fact, more generalized models such as the Rician162.4. Problem Formulationfading model would require more knowledge about the specific propagation environment.In particular, the Rician K-factor, which is the power ratio between the fixed and scattercomponents, is itself a random variable that varies with time, frequency and the location[57].For convenience, let us express the power allocation of cell c by pc = (pu,n)u∈Uc, n∈S ∈RN |Uc|+ and those of the other cells by the vector p−c , (pj)j∈U\Uc, n∈S ∈ RN(|U|−|Uc|)+ . Thesum rate of the cell c is thus given by:Rc (pc,p−c) =WN∑u∈UcN∑n=1log (1 + SINRu,n)where W is the total system bandwidth.By adopting the linear energy consumption model proposed in [58], the overall powercost for cell c is given byCc (pc) = Pf,c + ψc∑u∈UcN∑n=1pu,nwhere Pf,c denotes the fixed power cost due to the BS circuitry and cooling equipment andψc is the BS power amplifier efficiency.2.4 Problem FormulationIn this chapter, we consider the general problem of maximizing the sum of energy efficien-cies of the two-tier cells subject to a set of power constraints:maximizep∑c∈C EEc (pc,p−c)subject to hl (p) ≤ 0, ∀l ∈ V .(2.2)172.4. Problem Formulationwhere each function hl, ∀l ∈ V is assumed to be affine. Here, the bit-per-Joule energyefficiency of each cell c is given by the ratio of its sum rate over its power cost:EEc (pc,p−c) =Rc (pc,p−c)Cc (pc). (2.3)In general, the sum energy efficiency is appropriate to measure the overall energy effi-ciency of two-tier networks assuming the BSs have independent transmissions and powersources. However, the following system energy efficiency metric can also be usedEE0 (p) =∑c∈C Rc (pc,p−c)Cc (pc)(2.4)when the small-cell base stations have a single power source or when they are able sharetheir energy.For example, in C-RAN heterogeneous networks, the small cells are served by remoteradio units which are connected to a central unit. In addition, the system energy efficiency(2.4) is also useful in a future scenario, in which the cells will be able to harvest energyfrom renewable sources and share their energy through a smart-grid infrastructure.Under the above assumptions, the optimization problem becomes:maximizepEE0 (p)subject to hl (p) ≤ 0, ∀l ∈ V .(2.5)Due to the interference, neither the sum energy efficiency (2.3) nor the system energyefficiency (2.4) is a concave function. Thus, the above problems cannot be directly solvedwith standard convex optimization algorithms. In addition, problem (2.2) is more compli-cated than problem (2.5) since the former maximizes a sum of non-convex fractions. Inthese optimization problems, we have the following power constraints1:1Note that not all these types of power constraints must be imposed. The constraints of the resourceallocation will depend on both the network architecture and deployment scenario.182.4. Problem FormulationIndividual transmit power constraintsDue to regulations and hardware limitations, the macrocell and small-cell base stationsare subject to transmit power limitations. For a standard cellular deployment, each basestation has its own power supply and is subject to an individual power constraint:∑u∈UcN∑n=1pu,n ≤ Pc, ∀c ∈ C.Total transmit power constraintFor the case of cloud-radio access network (C-RAN) deployment, the two-tier cells areserved by RRUs and may share a total transmit power constraint. In a C-RAN, the lowpower requirements of RRUs can enable them to be powered by a centralized DC powersupply that is located at the central unit [28]. In that case, the cells may share a totaltransmit power constraint:∑c∈C∑u∈UcN∑n=1pu,n ≤ Ptotal.Interference temperature constraintsIt is also possible for future two-tier cells to use a spectrum band that is licensed to aprimary system. Using a cognitive underlay model, the primary and secondary systems co-exist and share the same spectrum. Assuming there are primary users V ={v1, . . . , v|V|},the two-tier cells should not exceed the interference power limit Imaxl measured in Wattsimposed by each primary user l as follows∑c∈C∑u∈UcN∑n=1pu,nglc,n ≤ Imaxl , ∀l ∈ V , (2.6)where glu,n is the channel gain on subcarrier n between a primary user l and the BS thatserves a user u. Note that it is also possible to adopt this cognitive underlay mechanismin the resource allocation of the heterogeneous network. Precisely, the small cells are192.4. Problem Formulationconsidered as secondary users whereas the macrocell users are the primary users. In fact,such spectrum sharing scheme has the benefits of mitigating the inter-cell interference andimprove the spectrum efficiency of heterogeneous networks [59, 60].Probabilistic interference constraintsThe aforementioned cognitive underlay scheme works well if the BSs are able to estimatethe channels to the primary users. In practice, the channel information can be imperfectdue to estimation errors. To take this uncertainty into account, the deterministic inter-ference constraints in (2.6) can be replaced by probablistic interference constraints of theform:Pr{∑c∈C∑u∈UcN∑n=1pu,nglc,n ≤ Imax}≥ 1− , ∀l ∈ V (2.7)In (2.7), glc,n represents the uncertain channel gain which depends on the random variableζlc,n in an affine fashion as:glc,n = ĝlc,n + ∆glc,nζlc,n (2.8)where ĝlc,n denotes the estimated channel gain and ∆glc,n is the absolute value of themaximum channel estimation error. The random error variables {ζlc,n} are assumed tohave mutually independent distributions with E [ζlc,n] = 0 and |ζlc,n| ≤ 1, ∀ (l, c, n). Inother words, we assume that the uncertainty region of the random variables ζ = (ζlc,n)∀l,c,nis given by:D = {ζ | ‖ζ‖∞ ≤ 1} (2.9)Note that this bounded uncertainty model is reasonable because, with a limited coor-dination between the primary and secondary systems, it is difficult for the secondary usersto obtain complete information about the probability distributions. Instead, the small-cellBSs can get estimate the error bounds on the channel estimation through empirical mea-surements. Although the probabilistic contraints (2.7) are non-convex, we will show inSection 2.7 to handle them.202.5. Special Case: Orthogonal Multi-cell Transmissions2.5 Special Case: Orthogonal Multi-cell TransmissionsIn this section, we first simplify the generic problem 2.2 by assuming orthogonal transmis-sions for the users. One way to ensure orthogonality is by scheduling the users on differentfrequency subcarriers. We assume that this is performed before the power allocation. Analternative way to enable orthogonal transmissions is spatial multiplexing scheme but thisrequires multiple antennas at the base stations so as to cancel the interference betweenthe concurrent transmissions.2.5.1 Parametric Convex ProgrammingWith orthogonal transmissions, there is no interference between the users at each subcar-rier. Thus, the rate function Rc of each cell c is concave and depends only on its powerallocation vector pc. Nonetheless, the energy efficiency maximization problem is still hardto solve since the objective function is a sum of quasiconcave functions.Let us write this problem in its epigraph form to obtain the equivalent formulation:(P )maximizep,θ∑c∈C θcsubject to Rc (pc)− θcCc (pc) ≥ 0, ∀c ∈ C,hl (p) ≤ 0, ∀l ∈ V ,(2.10)in which we introduced some auxilliary variables θ , (θc)c∈C ∈ R|C|. To further simplify thenotation, let us gather the convex power constraints into a convex set K of feasible powerallocation as follows:K , p ∈ RN |U|×1 : hl (p) ≤ 0, ∀l ∈ V. (2.11)212.5. Special Case: Orthogonal Multi-cell TransmissionsNext, we introduce the following family of parametric convex problems (Px)(Px)maximizep∑c∈C αc (Rc (pc)− θcCc (pc))subject to p ∈ K,(2.12)which is parametrized by the following vector of parameters x , (α, θ) ∈ R2|C| with α ,(αc)c∈U ∈ R|C|. Note that whereas the vector θ corresponds to optimization variables in theoriginal problem (P ), it is considered as a vector of parameters in (Px). We can interpretthe parametric problem (Px) as a weighted sum rate maximization with penalty terms.Precisely, α is a priority weight vector and θ is a pricing vector for the penalty terms.Since (Px) is convex, its solution can be efficiently and reliably computed using standardalgorithms [61]. In our simulations, we employed the disciplined convex optimizationsoftware CVX to solve the convex subproblem (2.12) [1].To find an easier way to solve the non-convex problem (P ), we establish the followingrelation between (P ) and the convex parametric problem (Px).Proposition 2.1. If(p̂, θ̂)is an optimal solution of (P ), then there exists α̂ such that p̂ is asolution of (Px) with x =(α̂, θ̂). Moreover, p̂ must satisfy the following equations:Rc (p̂c)− θ̂cCc (p̂c) = 0, ∀c ∈ C, (2.13)α̂cCc (p̂c)− 1 = 0, ∀c ∈ C. (2.14)Proof. See Appendix A.1.Since the energy maximization problem (P ) in (2.10) is always feasible, the aboveresults means that there exists at least one parameter x that satisfies the following twoconditions. First, the solution p̂ of the corresponding convex parametric problem (Px)coincides with the optimal solution of (P ). Second, the solution p̂ must satisfy the systemof 2 |C| nonlinear equations in (2.13) and (2.14).222.5. Special Case: Orthogonal Multi-cell TransmissionsFor conciseness, let us define a function p˜ (x) : R2|C| → K which takes as input a pa-rameter x and returns an optimal solution of the parametric problem (Px). Mathematically,we have:p˜ (x) = argmaxhl(p)≤0, ∀l∈V∑c∈Cαc (Rc (pc)− θcCc (pc)) (2.15)Then, we also introduce the following system of nonlinear equations:F (x) = 0, (2.16)where the components of the vector-valued function F : R2|C| → R2|C| are defined by:Fc (x) = θcCc (p˜ (x))−Rc (p˜ (x)) , ∀c ∈ C (2.17)Fc+|C| (x) = αcCc (p˜ (x))− 1, ∀c ∈ C. (2.18)Each of these component functions is defined by composite functions of the cell rate Rc orthe power cost Cc with the function p˜ defined in (2.15).Given the above definitions, Proposition 2.1 equivalently says the following. If a powerallocation p̂ solves the energy maximization problem (P ), then there exists a parameterx =(α̂, θ̂)such that p̂ = p˜ (x) and the parameter x is a root of the system of nonlinearequations F (x) = 0. This indicates an indirect route for solving the non-convex optimiza-tion (P ). When (P ) is feasible, then at least one root of F (x) = 0 exists. Furthermore,if F (x) = 0 has a unique solution x̂, then a global optimal solution of the sum energymaximization problem (P ) is exactly given by p˜ (x̂). In the next part, we show that thesolution of F (x) = 0 is indeed unique. As a result, it is possible to solve (P ) by finding theparameter satisfying F (x) = 0 using an iterative method.2.5.2 Uniqueness of the Root of F (x) = 0To establish the uniqueness of the solution of F (x) = 0, we derive some useful propertiesof F (x). First, let us define X as the convex and compact set of parameters:232.5. Special Case: Orthogonal Multi-cell TransmissionsX ,{x = (α, θ) ∈ R2|U|+ | −αminc ≤ αc ≤ αmaxc , −θminc ≤ θc ≤ θmaxc}. (2.19)The bounds in (2.19) are defined for each cell c ∈ C as:αminc = minp∈K1Cc (pc), αmaxc = maxp∈K1Cc (pc), (2.20)θminc = minp∈KRc (pc)Cc (pc), θmaxc = maxp∈KRc (pc)Cc (pc). (2.21)These bounds are finite and well-defined whenever the feasible set K in (2.11) is non-empty.By definition, any solution of F (x) = 0 has to satisfy the conditions (2.13) and (2.14).Thus, given the bounds of X in (2.20) and (2.21), we can restrict the function F (x) tobe defined over the set X . Effectively, any solution of F (x) = 0 cannot lie outside of X .Then, we can obtain the following key properties for the function F (x).Lemma 2.1. When defined over the set X , the vector function F defined in (2.17) and (2.18)must satisfy the following:1. F is differentiable and its Jacobian F′ is a non-singular, diagonal and positive definitematrix given by:F′ (x) =diag((Cc (p˜ (x)))c∈C)0|C|×|C|0|C|×|C| diag((Cc (p˜ (x)))c∈C). (2.22)2. F is Lipschitz continuous and strongly monotone in X with constants M ≥ 0 andm > 0 respectively.3. The Jacobian F′ of F is Lipschitz continuous and its inverse is bounded.Proof. The proofs are given in Appendix A.2.242.5. Special Case: Orthogonal Multi-cell TransmissionsWith the above properties of F, we have the following useful result about the systemF (x) = 0 (2.16).Proposition 2.2. There is a unique solution to the nonlinear system F (x) = 0 in the set X .Proof. The proof is given in Appendix A.3.Consequently, if we can find the unique parameter x̂ that satisfies F (x) = 0, the optimalsolution of the energy efficiency maximization problem (P ) can be determined by solvingthe convex problem (Px) with its parameter set to x = x̂.2.5.3 Damped Newton AlgorithmGiven the results of the previous section, we derive an iterative algorithm for solving prob-lem (P ) using a damped Newton method [62]. The proposed algorithm is listed in Algo-rithm 2.1 on the next page. First, we initialize the power allocation p(0) ∈ K of the usersand set the initial parameter x(0) =(α(0), θ(0))as follows:α(0)c =1Cc(p(0)c) , ∀c ∈ C, (2.23)θ(0)c =Rc(p(0)c)Cc(p(0)c) , ∀c ∈ C. (2.24)In each iteration i, we solve the convex parametric subproblem in (2.12) for a givenparameter x(i) to obtain a new power allocation p˜(x(i)). After this, if the terminationcondition∥∥F(x(i))∥∥ <  is satisfied for a given tolerance  > 0, then we stop the algorithm.Otherwise, we update the parameter x(i) using a damped Newton method. Precisely, wefirst calculate the Newton step at the i-th iteration by:d(i) = −F′(x(i))−1F(x(i)), (2.25)252.5. Special Case: Orthogonal Multi-cell Transmissionswhere F′ is the Jacobian matrix in (2.22) of the vector-valued function F. From Lemma2.1, F′ (x) is a diagonal positive definite matrix; hence, its inverse can be easily computed.Then, we use a line search to control the step size 0 < λ < 1 so as to avoid overshooting[63]. In other words, we start with a full Newton step λ = 1 and reduce it by a factor0 < σ < 1 until a sufficient decrease in the norm of∥∥F(x(i) + λd(i))∥∥ is obtained [62].After this line search, we update the parameter vector as:x(i+1) , x(i) + λd(i). (2.26)By substituting (2.17), (2.18) and (2.22) into (2.25), the update equation (2.26) of theparameter x(i) =(α(i), θ(i))in step (S.4) is given component-wise by:α(i+1)c = (1− λ)α(i)c + λ1Cc(p(i)c) , ∀c ∈ C, (2.27)θ(i+1)i = (1− λ) θ(i)i + λRc(p(i)c)Cc(p(i)c) , ∀c ∈ C. (2.28)At the next iteration, we solve again the parametric problem (2.12) with the new pa-rameter x(i+1). This iterative process is carried on until convergence.Before presenting the convergence of this algoritm in the following proposition, weshould discuss the importance of the parameter updates in (2.27) and (2.28). As men-tioned earlier, the parameter αc plays the role of a priority weight in the parametric prob-lem (Px) whereas θc is a pricing factor that corresponds to the power penalty term of cellc. Intuititely, the priority weight α(i+1)c for the next iteration is calculated as a weightedsum of the most recent value α(i)c and the inverse of the power cost Cc(p(i)c). The pricingfactor θ(i+1)i is similarly updated based on the energy efficiency achieved by cell c in thelast iteration.Proposition 2.3. The iterative damped Newton Algorithm 2.1 converges to an optimal solu-tion of the power allocation problem (P ) in (2.10).262.6. General Case: Non-orthogonal TransmissionsAlgorithm 2.1: Algorithmic procedure for solving (P )Initialize: , 0 < (µ, λ, σ) < 1, p(0) ∈ RN |U|.Set x(0) using (2.23) and (2.24) and set i = 0,Repeat(S.1) : Compute p˜(x(i))by solving the parametric problem (2.12),(S.2) : Initialize λ← 1,(S.3) : Compute x(i) + λd(i) using (2.27)-(2.28),(S.4) : while∥∥F(x(i) + λd(i))∥∥ > (1− µλ)∥∥F(x(i))∥∥ do λ← σλ(S.5) : Update x(i+1) ← x(i) + λd(i)Until∥∥F(x(i))∥∥ ≤ Proof. According to Proposition 2.1, an optimal solution of (P ) must also be a solution ofa parametric problem (Px) whose parameter x is necessarily a solution of F (x) = 0. InProposition 2.2, we have shown that F (x) = 0 has a unique solution. Thus, Algorithm 2.1can retrieve an optimal solution of (P ) by iteratively finding the unique solution of F (x) =0 and solving the parametric convex problem (Px). A rigorous convergence analysis of theNewton method is available in [62, 63].2.6 General Case: Non-orthogonal TransmissionsIn this section, we tackle the energy efficiency maximization problem in (2.2) for thegeneral scenario. In other words, we relax the assumption that the transmissions areorthogonal. In other words, the cells are now assumed to fully reuse the spectrum and mayinterfere with each other. To tackle the difficulty in solving this optimization problem, weapply the principle of minorization-maximization in conjunction with the previous Newtonmethod. First, let us give the definition of a minorization.Definition 2.1. Given a function f defined over K ⊂ Rn, another function g defined overK ×K is said to minorize f if the following conditions are satisfied:f (p) ≥ g (p,q) ∀ (p,q) ∈ K ×K, (2.29)f (p) = g (p,p) ∀p ∈ K. (2.30)272.6. General Case: Non-orthogonal TransmissionsIn other words, g as a function of p and q is upper-bounded by f and coincides with f whenq = p. This property is very useful when we want to maximize a complicated function fand have access to a simpler minorizing function g. Instead of solving f directly, we usethe following sequential procedure to solve the original problem:p(j+1) = argmaxp∈Kg(p,p(j)), (2.31)where at each iteration j, we compute p(j+1) by maximizing the minorization function gwith q = p(j). The minorization-maximization (MM) is a general principle for solvingnon-convex optimization problems. For example, it has been widely used in statistics andmachine learning [64]. It has also surfaced in wireless resource allocation and localizationproblems in [65–67]. In these cited works, the utility function consists of a differenceof convex (D.C) functions; and the MM principle directly leads to the convex-concaveprocedure (CCCP) [68], a special algorithm for D.C programs. In contrast, our problemdoes not have this form so that the CCCP algorithm is not directly applicable.2.6.1 Problem ReformulationAs mentioned previously, the sum energy efficiency function is not convex because of theinterference coupling in the SINR expression and of its sum-fractional form. Therefore, itis useful to find a minorizing function of the energy efficiency utility function.For convenience, let us define for each user u ∈ U the following two functions:vu (p−u) =N∑n=1logσ2u,n +∑k∈U\{u}pk,ngk,n , (2.32)lu (p−u,q−u) = vu (q−u) +∇vu (q−u)> (p−u − q−u) , (2.33)where p−u ∈ RN(|U|−1) is the power allocation of all users except user u. The gradient ofvu (q−u) in (2.33) is a vector of length N (|U| − 1) given by282.6. General Case: Non-orthogonal Transmissions∇vu (q−u) =guj,nσ2u,n +∑k∈U\{u}pkgikj∈U\{u}, n∈S. (2.34)Then, we have the following result which provides a way to solve the energy efficiencymaximization (2.2) in its general form.Lemma 2.2. A minorization function for the sum energy efficiency function is given byg (p,q) ,WN∑c∈C∑u∈Uc[∑Nn=1 log(σ2u,n +∑j∈Upj,nguj,n)− lu (p−u,q−u)]Cc (pc). (2.35)Proof. The sum energy efficiency function (2.3) can be rewritten asf (p) =∑c∈C∑u∈Uc[∑Nn=1 log(σ2u,n +∑j∈Upjguj)− vu (p−u)]Cc (pc),where the function vu defined in (2.32) is a concave function of p−u. Therefore, it isbounded above by its first-order Taylor approximation lu defined in (2.33) asvu (p−u) ≤ lu (p−u,q−u) , ∀p−u,q−u, ∀u ∈ U (2.36)and the equality holds at q−u. Furthermore, since the power cost Cc (pc) is always apositive function, then we can deduce that:g (p,q) =WN∑c∈C∑u∈Uc[∑Nn=1 log(σ2u,n +∑j∈Upj,nguj,n)− lu (p−u,q−u)]Cc (pc)≤ f (p) . (2.37)In other words, the function g (p,q) defined in (2.35) minorizes the sum energy efficiencyfunction f (p) and the equality holds when q = p.The benefit of using g as the objective function of the surrogate maximization prob-lem (2.31) is that we can compute its global optimal solution using the damped Newton292.6. General Case: Non-orthogonal TransmissionsAlgorithm 2.2: Minorization-Maximization algorithm based on Newton methodInitialize: ,p(0) ∈ RN |U| and set j = 0.Repeat(S.1) : Minorize the EE utility function f at p(j) using (2.35)(S.2) : Update the power allocation as p(j+1) = argmaxx∈Kg(p,p(j))(S.3) : j ← j + 1Untilf(p(j+1))−f(p(j))f(p(j))≤ method presented in Section 2.5. This is explained in more details in the next subsection.2.6.2 Minorization-Maximization AlgorithmWe derive an iterative algorithm for the sum energy maximization problem for the case ofnon-orthogonal transmissions using the minorization-maximization (MM) procedure. Ateach iteration j, a minorization function g(p,p(j))of the sum energy efficiency is obtainedusing equation (2.35). Then, we maximize this surrogate function g(p,p(j))using thedamped Newton method presented in Section 2.5. Each solution of this subproblem givesus a new power allocation p(j+1) which is used to provide a new minorizing functiong(p,p(j+1))for the next iteration. This algorithm, which is outlined in Algorithm 2.2, hasa nice convergence property as stated by the following proposition.Proposition 2.4. The minorization-maximization algorithm monotonically converges to atleast a local optimum of the sum energy efficiency maximization problem.Proof. To show that the algorithm is monotonically convergent, we prove that:f(p(j+1))≥ f(p(j)), ∀j. (2.38)According to Lemma 2.2, g minorizes the energy efficiency function f . Thus by (2.29), wehave:f(p(j+1))≥ g(p(j+1),p(j)). (2.39)Next, we show that the damped Newton method can be used to compute:302.6. General Case: Non-orthogonal Transmissionsp(j+1) = argmaxx∈Kg(p,p(j)). (2.40)Using (2.35), the minorization maximization problem (2.40) can be written in an epigraphform: maximizep,θ˜∑c∈C θ˜csubject to R˜c (p)− θ˜cCc (pc) ≥ 0, ∀c ∈ C,hl (p) ≤ 0, ∀l ∈ V ,(2.41)where θ˜ are new auxilliary variables.The minorized functions R˜c in (2.41) are given byR˜c (p) ,∑u∈Uc[N∑n=1log(σ2u,n +∑j∈Upj,nguj,n)− lu (p−u,q−u)], ∀c ∈ C.In contrast to the cell sum rate function, each minorized function R˜c is strictly concave inp. As a result, the surrogate problem in (2.40) share the same properties as the problem(P ) in (2.10). This means that we can also use the convex parametric approach presentedin Section 2.5 to solve (2.40). Hence, the statements in Propositions 2.1 and 2.2 also holdfor (2.40). Thus, the damped Newton method in Algorithm 2.1 can be used to find thepower allocation p(j+1) that maximizes the function g(p,p(j)).Therefore, the following holds at the j-th iteration of Algorithm 2.2:g(p(j+1),p(j))≥ g(p(j),p(j))= f(p(j)), (2.42)where the last equality is due to the definition of minorization function in (2.30). Bycombining (2.39) and (2.42), we proved that f(p(j+1))≥ f(p(j)), ∀j.Even if the original problem is not convex, our proposed algorithm can produce anefficient power allocation using the minorization-maximization procedure and Newton312.6. General Case: Non-orthogonal Transmissionsmethod. In the next section, we verify the convergence properties and analyze the per-formance of these algorithms.2.6.3 Practical Issues for ImplementationGiven the above-proposed algorithm, we should discuss two aspects that are important forimplementation.First, the proposed resource allocation scheme relies on accurate channel state esti-mates. This downlink channel estimation can be done through pilot signal training eitherdirectly from the base stations to the mobile users in a Frequency Division Duplex system(FDD) or reversely from the mobile users to the base stations in a Time Division Duplexsystem (TDD). The later is preferable assuming the channel reciprocity is preserved in thesmall-cell propagation environment. In fact, the direct channel estimation requires a feed-back link for the users to communicate the channel estimates to the base stations. Thismay be expensive in terms of spectrum efficiency especially since it has to be done overthe air. Exploiting channel reciprocity in a TDD system, we can avoid this overhead. Whilewe assume perfect channel information between the base stations and served users of thetwo-tier network, it is more difficult to acquire perfect channel information for the channeltowards primary users due to the lack of coordination. In this case, we explicitly considerthe channel uncertainty that may be present in the interference constraints in the nextsection.Second, it is also crucial that the resource allocation must satisfy a delay requirementthat does not exceed the channel coherence time. This coherence time approximatelyequals to the inverse of Doppler spread and depends on the mobility of the users. Wecan envisage that small-cells are mostly deployed in urban areas where most communi-cation happens indoor. In such case, the coherence time can be within 500 − 3000ms ata 2.5Ghz frequency band [69]. Therefore, the channel state information must be sharedby the base stations to the central unit before the channel state changes. To meet this re-322.7. Power Allocation with Probabilistic Interference Constraintsquirement, high-capacity front-haul links are able to meet such demand. In fact, fronhaullink requirements, as specified by mobile network operators, should have a delay in theorder of 45-250ms with a maximum supporting distance between the RRU and CU of 50km[70]. Our proposed method can implemented by solving the inner problem with standardconvex optimization algorithms, such as interior-point methods which have a polynomial-time complexity [61]. As shown in our simulation results, the algorithm converges veryfast, within 10 iterations. Therefore, it is practical to be used in the presented centralizedframework since the central unit has a much higher processing power than a standard basestation [71, 72].2.7 Power Allocation with Probabilistic InterferenceConstraintsIn the following, we consider the energy efficient resource allocation with probablistic in-terference power constraints due to the channel uncertainty between the primary usersand the small-cell BSs. To handle the intractability of the probabilistic constraints (2.7),we replace them by robust approximation framework [73]. In other words, we replace theprobabilistic constraints with conservative yet convex ones. Denote the concatenation ofchannel uncertainties for each primary user l by the vector ζl , (ζlc,n)c∈C,n∈S . By substitut-ing the channel gain (2.8), we can rewrite (2.7) for each c as:Pr{∑c∈C∑u∈UcN∑n=1∆glc,npu,nζlc,n ≤ Imax −∑c∈C∑u∈UcN∑n=1pu,nĝlc,n}≥ 1−  (2.43)332.7. Power Allocation with Probabilistic Interference ConstraintsBall-box approximationThe above constraint (2.7) can be replaced by the ball-box approximation given by [73]:∑u∈Upun (ĝlc,n + ∆glc,nζlc,n) ≤ Imax, ∀ζl ∈ Z1 (2.44)Z1 , {ζl | ‖ζl‖2 ≤ Ω, ‖ζl‖∞ ≤ } (2.45)which is equivalent to the following system of constraints:zlc,n + ulc,n = −∆glc,npu,n, ∀u ∈ Uc, ∀c ∈ C (2.46)∑c∈C∑u∈UcN∑n=1|zlc,n|+ Ω√√√√∑c∈C∑u∈UcN∑n=1u2lc,n ≤ Imax −∑c∈C∑u∈UcN∑n=1pu,nĝlc,n (2.47)where {zilc}i and {uilc}i are new slack variables. The following proposition, which is dueto [73], states that we can use the above constraints to replace the probabilistic constraints(2.43).Proposition 2.5. [73] Every feasible power allocation p of the system of constraints (2.46)-(2.47) also satisfies the probabilistic interference constraint (2.43) provided that Ω ≥√2 ln (1/).Thus, all feasible power allocations with respect to the above constraints (2.46) and(2.47) satisfy the interference power constraint∑c∈C∑u∈Uc∑Nn=1 pu,n (ĝlc,n + ∆glc,nζlc,n) ≤Imax with a probability of at least 1− .In contrast to the probabilistic constraint (2.43), constraints (2.46) are more tractablesince they are represented by convex conic constraints. Interestingly, constraint (2.46) canalso be interpreted as a robust interference constraint in which the channel uncertaintybelongs to the intersection of a ball with radius Ω and a box with an edge length of 2 bothcentered at the origin.342.7. Power Allocation with Probabilistic Interference ConstraintsBudgeted robust approximationAlternatively, we also consider the budgeted robust approximation of the probabilistic con-straints (2.43) defined by:∑u∈Upun (ĝlc,n + ∆glc,nζlc,n) ≤ Imax, ∀ζl ∈ Z2 (2.48)Z2 ,{ζl | ‖ζl‖∞ ≤ ,∑c∈C∑u∈UcN∑n=|ζlc,n| ≤ γ}(2.49)It can also be shown that the above budgeted robust constraints (2.43) can be equiva-lently represented by the following finite system of linear constraints [73]:zlc,n + ulc,n = −∆glc,npu,n, ∀u ∈ Uc, ∀c ∈ C (2.50)∑c∈C∑u∈UcN∑n=1|zlc,n|+ γmaxc∈C|ulc,n| ≤ Imax −∑c∈C∑u∈UcN∑n=1pu,nĝlc,n (2.51)Then, the following proposition gives the value of the budget parameter γ that guaran-tees the satisfaction of the probabilistic constraints (2.43) [73].Proposition 2.6. [73] Let (pi)i∈U be a feasible solution of the system of constraints (2.50)-(2.51), then it also satisfies the probabilistic interference constraint (2.43) provided that weset γ ≥√2 |U| · |C| ln (1/).Compared to the ball-box approximation (2.46), the budgeted approximation is moreconservative when the uncertainty budget parameter γ is linked to the safety parameterΩ in the ball-box approximation according to γ = Ω√|U| · |C|. However, the advantage ofthe budgeted approximation is that we could model the robustness with linear constraints,thus reducing the computational complexity especially if one wants to decompose theoptimization problem.352.8. Simulation ResultsWorst-case robust interference constraintsFinally, it is possible to consider a worst-case robust approach by approximating the inter-ference constraints with:∑c∈C∑u∈UcN∑n=1pu,n (ĝlc,n + ∆glc,nζlc,n) ≤ Imax, ∀ζl∈ Dl (2.52)Dl , {ζl | −1 ≤ ζil,c ≤ 1, ∀ (i, c) ∈ U × C} (2.53)This is also equivalent to:∑c∈C∑u∈UcN∑n=1|∆glc,npu,n| ≤ Imax −∑c∈C∑u∈UcN∑n=1pu,nĝlc,n (2.54)Consequently, the energy efficiency optimization framework can also handle probabilis-tic power constrains using the robust approximation framework presented above.2.8 Simulation ResultsIn this section, we present our simulation results for a two-tier wireless system composedof M primary users served by the macrocell and N secondary users, each of which isserved by a hotspot picocell BS. Based on the WINNER models [74], the large-scale pathloss attenuation PL (in dB) is calculated as a function of the distance (in meters) andcarrier frequency (fc in GHz):PL = A log10 (d) +B + C log10(fc5.0)+ S [dB] (2.55)where S is the random log-normal shadowing. For the two-tier cellular network, the modelparameters for each respective scenario are detailed in Table 2.1. The maximum values oftotal transmit powers and the small-cell BSs’ circuit-power consumption are also listed in362.8. Simulation ResultsTable 2.1 – Simulation parameters in Chapter 2.Parameters ValuesCarrier frequency fc = 1.9 GHzSystem bandwidth 5 MHzSub-channel bandwidth 180 kHzNumber of primary macro users 5SBS-to-SU path loss (model A1-LOS) A = 1.87, B = 46.8, C = 20SBS-to-PU path loss (model A1-NLOS) A = 3.68, B = 43.8, C = 20MBS-to-SU path loss (model C1-NLOS) A = 3.36, B = 44.36, C = 23MBS antenna height hMBS = 50 mStandard deviation of SBS-to-SU shadowing S = 3 dBStandard deviation of SBS-to-PU shadowing S = 4 dBStandard deviation of MBS-to-SU shadowing S = 6 dBMBS maximum transmit power 46 dBmSBS maximum transmit power 36 dBmSBS circuit power consumption 20 WNoise Power Density −174 dBm/Hz372.8. Simulation ResultsTable 2.1. Since we consider only a transmission on one sub-channel, we normalized thesevalues in accordance with the subchannel bandwidth.With uniform distribution, we randomly placed N small-cell BS in a 1km by 1km gridregion, at the center of which the macrocell BS is located. The optimization of small-celltransmissions follows after the cell association. Using a simple weighted max-SNR associa-tion rule, we determined the macrocell and small-cell coverage regions as multiplicativelyweighted Voronoi tesselations [75]. The weigths were specified by the effective cell asso-ciation bias β between the macrocell and small cells which can be controlled to expandor shrink the small-cells’ range. Precisely, β is a weighted ratio of the received SNR frommacrocell BS versus small-cell BS [76]. For simplifying our analysis, we place all small-cellusers on their cell edges as illustrated in Fig. 2.2.2.8.1 Convergence ResultsFirst, we study the convergence of the proposed algorithms for the case of orthogonaltransmissions and Imax = 0dB with respect to noise level. For simplicity, we set the in-terference power limit Imax to be equal for every macrocell user. In Fig. 2.3a, a fastconvergence is observed for the damped Newton algorithm. Its convergence is displayedin terms of ‖F (x)‖. We see that the optimal parameter x∗ can be found within 5 iterationsfor a specified tolerance  = 10−4.For the general case, we employ Algorithm 2.2 to find an energy-efficient power alloca-tion. Fig. 2.3b shows that it achieves near-optimal performance within a total of 10 inneriterations.2.8.2 Performance BenchmarkIn the following, we compare the average performance of the proposed spectrum sharingscheme with that of the algorithm in [56], which is based on non-cooperative game theory.382.8. Simulation ResultsEective cell association bias:= 9dB= 6dBMacrocell base station Small-cell base stationMacrocell user Small-cell userFigure 2.2 – Cell association regions of a two-tier network with 5 macro-cell users and 10 small-cell BSs.Solid and dashed circular lines show the cell edges with effective cell association bias β of 6 dB and 9 dBrespectively. Small-cell users are randomly located on the cell edges.392.8. Simulation Results1 2 3 4 5 610−710−610−510−410−310−210−1IterationNorm   Fxn () ()(a) Damped Netwon method with N = 3, M = 2.0 10 20 30 40 50 60 70 8000.20.40.60.81Overall iteration indexEnergy efficiency (bit/J/Hz)    Proposed scheme (  ε = 10−6 )  Proposed scheme (  ε = 10−4 )  Non−cooperative game(b) Minorization-Maximization algorithm with N = 10and M = 5.Figure 2.3 – Convergence of proposed algorithms.The comparison is made for different number of small cells N and for different values ofthe cell association bias β and of the interference limit Imax (in dB) relative to the noisepower. For a given set of simulation parameters (N, β, Imax), we generate 200 randominstances of the network topology and channel realization, over which we average theperformance of the proposed and comparative schemes.First, the average sum energy efficiency is shown in Fig. 2.4. It is seen that our proposedscheme outperforms the non-cooperative approach. When an optimal power allocation isemployed, the energy efficiency increases with the number of active small cells. In contrast,the performance achieved by the Nash equilibrium solution decreases when more smallcells compete in a selfish way. This result is also reflected in the convergence behavior ofthe non-cooperative game as shown in Fig. 2.3b. Furthermore, Fig. 2.4 shows that higherenergy efficiency can be obtained when small-cell uses shorter cell range. This correspondsto β = 9dB in our example. Next, Fig. 2.5 shows that our proposed scheme provides abetter sum rate performance compared to non-cooperative scheme, especially when thesmall cells’ range are bigger, i.e. with β = 6dB.402.8. Simulation Results5 10 15 20 25 3000.511.52Number of Active Small CellsSum energy efficiency of small cells (bit/J/Hz)    Proposed scheme (β = 9dB, Imax = 20dB)  Proposed scheme (β = 9dB, Imax = 10dB)  Proposed scheme (β = 6dB, Imax = 20dB)  Proposed scheme (β = 6dB, Imax = 10dB)  Non−cooperative game (β = 9dB)  Non−cooperative game (β = 6dB)Figure 2.4 – Average sum energy efficiency comparison for different number of small cells.412.8. Simulation Results5 10 15 20 25 300510152025Number of active small cellsSum rate of small cells (bit/s/Hz)    Proposed scheme (β = 9dB, Imax = 20dB)  Proposed scheme (β = 9dB, Imax = 10dB)  Proposed scheme (β = 6dB, Imax = 20dB)  Proposed scheme (β = 6dB, Imax = 10dB)  Non−cooperative game (β = 9dB)  Non−cooperative game (β = 6dB)= 9dB= 6dBFigure 2.5 – Average sum rate comparison for different number of small cells.422.8. Simulation Results5 10 15 20 25 300102030405060Number of active small cellsAverage total interference at a primary user (dB)    Proposed scheme (β = 9dB, Imax = 20dB)  Proposed scheme (β = 9dB, Imax = 10dB)  Proposed scheme (β = 6dB, Imax = 20dB)  Proposed scheme (β = 6dB, Imax = 10dB)  Non−cooperative game (β = 9dB)  Non−cooperative game (β = 6dB)Imax = 20dBImax = 10dBFigure 2.6 – Average total interference power at a macro-cell user.As we expected, the performance of our scheme slightly decreases when the interfer-ence limit imposed by the macrocell users is stringent. For instance, when Imax = 10dBand β = 9dB, it appears that the non-cooperative scheme provides a higher sum rate thanour proposed scheme. Nonetheless, the comparison is not fair since the non-cooperativescheme ignores the interference power limits imposed by the macrocell users. Indeed,Fig. 2.6 shows that with the non-cooperative scheme, the interference power received bythe primary users always exceed the limit imposed by up to 40dB margin when there aremany active small-cells in the two-tier network. In contrast, our proposed scheme alwayssatisfies the total interference limit imposed by the primary users.432.8. Simulation Results−20 −15 −10 −5 0 5 10 15 20051015Interference Temperature Limit (dB)Sum Rate (bps/Hz)  00.050.10.15Energy efficiency (bps/Hz/J)Sum Rate − Perfect CSISum Rate − Worst caseSum Rate − Ball Box approxSum Rate − Budgeted approxEE − Perfect CSIEE − Worst caseEE − Ball Box approxEE − Budgeted approxFigure 2.7 – Performance of sum rate-optimal power allocation vs. Imax with P0 = 10 dB.2.8.3 Impact of Imperfect Channel InformationHere, we simulate the resource allocation with probabilistic interference constraints fora system of 3 small-cells and 2 primary users. We use |C| = 64 subcarriers. For theprobabilistic interference constraints, we set the threshold to  = 0.1. For simplicity, weassume that the maximum channel estimation error is set to ∆glc,n = 0.7× ĝlc,n.First, we plot in Figure 2.7 the performance of the rate-maximizing power allocation.We see that the optimal sum rate increases to infinity with the allowable interferencetemperature limit. In contrast, the resulting energy efficiency reaches a maximum atI thresmax = −4dB. Increasing the power beyond this threshold will largely reduce the energyefficiency.In Figure 2.8, we show the performance of the energy-efficiency maximizing powerallocation. In accordance with the previous observation, we see that the energy-efficiencyand the corresponding sum rate saturate when the interference limit is larger than a cer-tain threshold I thresmax . Beyond this threshold, the performance of the different robust power442.8. Simulation Results−20 −15 −10 −5 0 5 10 15 2000.020.040.060.080.1Interference temperature limit (dB)Energy efficiency (bps/Hz/J)  012345Sum Rate (bps/Hz)EE − Perfect CSIEE − Worst caseEE − Ball Box approxEE − Budgeted approxSum Rate − Perfect CSISum Rate − Worst caseSum Rate − Ball Box approxSum Rate − Budgeted approx(a) Fixed power P0 = 10−20 −15 −10 −5 0 5 10 15 2000.020.040.060.080.1Interference temperature limit (dB)Energy efficiency (bps/Hz/J)  012345Sum Rate (bps/Hz)EE − Perfect CSIEE − Worst caseEE − Ball Box approxEE − Budgeted approxSum Rate − Perfect CSISum Rate − Worst caseSum Rate − Ball Box approxSum Rate − Budgeted approx(b) Fixed power P0 = 100Figure 2.8 – Performance of energy-efficient optimal power allocation vs. Imax for different P0.452.8. Simulation Resultsallocations are equal to that of perfect channel state information (CSI). This is because themobile users keep their power transmission at the same level in order to avoid reducingthe energy efficiency. Therefore, when Imax > Ithresmax the energy-efficient transmission is au-tomatically robust against channel uncertainty. When Imax ≤ Ithresmax , we can observe that theworst-case robust power allocation has the lowest performance and that the budgeted ap-proximation is more conservative than the ball-box approximation. Comparison betweenfigures 2.8(a) and 2.8(b) show that the optimal energy efficiency decreases when the fixedpower cost P0 increases. However, the mobile users can achieve a higher sum rate up to5 bps /Hz when P0 = 20 dB and Imax = 5dB. These results illustrate the crucial trade-offbetween spectrum and energy efficiencies.46Chapter 3Energy Allocation and Cooperation forHybrid-Powered Two-tier Networks3.1 IntroductionIn this chapter, we investigate the joint energy allocation and cooperation scheme forwireless two-tier networks. First, we study the energy-efficiency optimization problemwith each cell having access to both a hybrid power source and an energy storage system.In other words, each base station is powered by both renewable sources and a conven-tional grid source. As previously mentioned, the large-scale deployment of small-cell basestations can pose a problem to the overall energy efficiency of multi-tier networks [8].Therefore, new sustainable energy solutions are also needed to increase the network ca-pacity in a cost-efficient manner.In Section 3.4, we study the energy-efficiency maximization problem with hybrid powersource. By utilizing hybrid source and energy storage, each cell can manage its energyover time based on the time-varying channel conditions and renewable energy arrivals.The objective is again to maximize the network energy efficiency by utilizing the availableenergy from the hybrid source in the most efficient manner over time. In doing so, wealso seek to satisfy an average rate constraint at each cell. While the main design targetis to improve the energy efficiency of two-tier networks, the spectrum sharing betweenmacrocell and small cells is also enforced because of the scarcity of wireless spectrum[77].473.2. Related WorksTo tackle the fluctuations of renewable energy arrivals, we also investigate the energycooperation in the multi-cell resource allocation problem in Section 3.5. Specifically, weassume that each cell can transfer some of its harvested energy to other cells through asmart-grid power infrastructure. In this chapter, we focus on offline optimization and as-sume a non-causal information about the channel and energy arrivals. In fact, our aimis first to analyze the theoretical benefits of using hybrid power source and energy coop-eration for wireless two-tier networks. In Section 3.6, we present numerical results andanalyze the potential gain of the proposed schemes. As such, the resource optimizationused in chapter is also performed in a centralized manner. For practical systems, it is how-ever important to design online and distributed algorithms that coordinate and adapt theenergy allocation between the cells over the time. For the design of such online algorithms,we later suggest some solution approaches in Section 5.2.3.2 Related WorksThe use of renewable energy and hybrid energy source for powering cellular networks is arelatively new idea. While many studies have considered pure energy-harvesting systems,only few have explored the use of hybrid power source and its implications. Previously,[10] studied the energy-efficient power allocation for a single-cell downlink system witha hybrid power source. However, [10] assumed an orthogonal access scheme and con-sidered only a single-cell system. Instead, we consider the multi-cell two-tier system withfull spectrum reuse. The idea of energy cooperation for cellular networks was initiallyproposed in [78] and [79] and is motivated by recent progress in renewable energy inte-gration and two-way energy flow in smart grid [42]. However, the study in [78] used asimplified model by abstracting the energy demand at each cell. On the other hand, [79]proposed joint power-spectrum allocation schemes but it assumed that the cells do notstore the harvested energy for future use. Instead, they optimized the energy cooperation483.3. System Modelassuming independent time frames. In contrast to the previous works, we simultaneouslyexploit the spatial and time diversities associated with the harvesting of renewable energyacross the cells in a dynamic fashion. When the cells have access to an energy storage,it is possible for them to dynamically allocate the stored energy depending on the timefluctuations of the energy harvesting. Similarly, energy sharing between the cells shouldhelp to mitigate the imbalance of energy arrivals across geographically separated cells.3.3 System ModelLet us consider the downlink of a two-tier multicell network which consists of M macrocellbase stations (MBSs) and K small-cell base stations (SBSs). Together, these base stationsserve a total of N users. Let us denote the set of macrocells and small cells in the networkby M and K respectively. Similarly, the set of users served by a macrocell m is definedby Um whereas that of a small-cell k is defined by Uk. Therefore, the set of all N usersis U ,(⋃m∈MUm)∪(⋃k∈KUk). We assume that all macrocells and small-cells share thespectrum. Although the resource allocation problem and algorithms studied in this chaptercan be readily applied to multi-carrier systems, we consider the transmission on a singlechannel in the following to simplify the exposition. The signal-to-interference-plus-noisepower ratio (SINR) of a macrocell or small-cell user u at the t-th time slot is given by:γu,t =guu,tpu,t∑j∈U\{u}guj,tpj,t + σ2u,t(3.1)where guu,t and and guj,t are the channel gains of user u at time t respectively from its BSserving and from an interfering BS that is serving another user j. Then, σ2u is the varianceof the AWGN noise at user u.In (3.1), the channel gains accounts for the large-scale path loss attenuation, the log-normal shadowing due to obstacles in the environment and the small-scale Rayleigh fad-493.3. System ModelTable 3.1 – List of key parameters in Chapter 3.Symbol Parameter descriptionM Number of macrocell base stationsK Number of small-cell base stationsN Total number of usersM Set of macrocellsK Set of small cellsc Cell index c ∈M∪KUc Set of users served by cell ct Time slot indexTs Time slot durationpu,t Power allocated to user u at time slot tguj,t Channel gain of user u at time t from the BS serving user jγu,t SINR of user u at time slot tpgu,t Power drawn from the conventional source for user u at time slot tphu,t Power drawn from the renewable source for user u at time slot tf Energy harvesting frame indexL Number of time slots in one frameNF Total number of framesHc,f Renewable energy harvested by cell c from the renewable source during frame fωc,f Excess power ωc,i discarded by cell c at the end of frame fBc Capacity of energy storage system at cell c−→h c,d,f Renewable energy transferred by cell c to cell d at the end of frame fη Energy transfer efficiency factorδc,i Net energy injected by a cell c into the aggregator at the end of frame i503.3. System Modeling. Furthermore, we assume that the small-scale fading is slow and frequency-flat. Thetime correlation of the fading samples is modeled using an autoregressive model accordingto Clarke’s scattering model [69, 80].Remark 3.1. The flat fading assumption holds when the maximum excess delay due to mul-tipath propagation is smaller than the symbol period of the transmitted signal. For mod-ern wireless systems, the use of orthogonal frequency division multiplexing technique helpreduce a wideband frequency-selective channel into parallel frequency-flat subchannelsgiven the subcarrier bandwidth is sufficiently small. In this case, the coherence bandwidthbecomes much larger than the subcarrier bandwidth.In this work, we assume that every base station receives its energy from the conven-tional power grid source and from a renewable energy-harvesting source. Moreover, abattery is used by each BS to store the harvested energy from the renewable source. Thisstored energy will then be used for future transmission. Here, we assume that the batteryhas no leakage loss because the timeframe for the considered power allocation problemis relatively short. In (3.1), the power allocated to all users at time slot t is denoted bythe vector pt = (pu,t)u∈U ∈ RN+ . Since each BS has a hybrid power source, we need todistinguish between the power pgt =(pgu,t)u∈U∈ RN+ drawn from the conventional powergrid and the transmit power pht =(phu,t)u∈U∪V∈ RN+ from the energy-harvesting source.These two power usage vectors must add up to satisfy the equation:pgu,t + phu,t = pu,t, ∀u ∈ U , ∀t. (3.2)Next, we describe the generic energy management process of the cells. In our model,we have two different time scales for managing the renewable energy and for allocatingthe transmission power. Precisely, each energy-harvesting frame is divided into L com-munication time slots. Each time slot has a duration Ts. During the i-th frame, each cellc ∈ M ∪ K harvests a total amount of renewable energy Hc,i from the renewable source.513.3. System ModelChannel fadingEnergy framesH0 H1 H2Hi Hi+1Communicationtime slotsHN10 2i i+1N#1 #2 #3 #(L 1)# L 2( )# l # LHi :i :p i 1( )L+lh :Total harvested power during energy timeframe i Discarded power at the end of energy timeframe iRenewable power allocated at time slot l of the frame ip i 1( )L+lhFigure 3.1 – Model for renewable energy management.At the end of the frame, some of this energy is stored into its battery, whose maximumcapacity is denoted by Bc. Then, the BS can allocate some of the stored renewable energyfor transmission during the next frame or the subsequent ones. In summary, the BSs man-age their harvested energy per energy-harvesting frame and allocate the stored energy forcommunication over the time slots. In our model, we consider a total transmit period ofT = LNF time slots where NF is the number of frames. This model is illustrated in Figure3.1.As explained above, each cell can store a part of the harvested energy in a battery forfuture use. To make sure that the cells can use only the energy that is available before thebeginning of each frame, the following energy causality constraints need to be satisfied:f∑i=1L∑l=1∑u∈Ucphu,(i−1)L+lTs ≤f−1∑i=0(Hc,i − ωc,i) , ∀c ∈M∪K, ∀f = 1, . . . , NF . (3.3)The left-hand side of this inequality is the amount of the energy that was used by each cellc for transmission during the first f frames. It should not exceed the available energy that523.3. System Modelwas accumulated up until the beginning of the f -th frame.The auxiliary variable ωc,i introduced in (3.3) is the excess energy that must be dis-carded by cell c at the end of the harvesting frame i. This is because cell c has a limitedbattery capacity Bc and a maximum transmit power limit Pc. As a consequence, it is pos-sible that cell c cannot store all the harvested power at the end of an energy-harvestingframe. This situation arises, for example, when the energy storage is already full at thebeginning of a frame and the cell still harvests a large amount of energy that exceeds itstransmit power limit Pc. Consequently, cell c has to discard the excess power ωc,i at theend of each frame i to satisfy its battery capacity constraints:f∑i=1(Hc,i −L∑l=1∑u∈Ucphu,(i−1)L+lTs − ωc,i)≤ Bc, ∀c ∈M∪K, ∀f = 1, . . . , NF − 1. (3.4)Given the above hybrid power source model, we employ the energy efficiency metricdefined by:EEc (p,pgc) =Rc (p)Cc (pgc), ∀c ∈M∪K.It is calculated as the ratio of its sum rate Rc (p) =∑u∈UcT∑t=1log(1 + γu,t)over its overallpower consumption Cc (pcc). Specifically, Cc is an affine function defined by [58]:Cc (pgc) = Pfc + ψT∑t=1∑u∈Ucpgu,t (3.5)where P fc denotes the additional power dissipation due to BS’s circuitry and cooling systemand ψ is the RF power amplifier efficiency respectively. In this model, the cost functionaccounts only for the energy drawn from the conventional power grid and considers therenewable energy as free. In addition, we assume that the circuit-power consumption P fcof each BS is powered by the conventional power grid source.533.4. Energy Allocation Problem3.4 Energy Allocation ProblemIn the following, we formulate the offline power allocation problem for the hybrid-poweredwireless two-tier systems. The first optimization variable is the vector of power allocationp = (pt)t=1,...,T ∈ RNT+ for all users. In addition, we optimize the following three powerusage vectors for the cells: (1) the transmit power pg = (pgt )t=1,...,T ∈ RNT+ used fromthe power grid, (2) the power ph =(pht)t=1,...,T∈ RNT+ used from the energy-harvestingsource, and (3) the renewable power wastage vector w = (ωc,f )f=1,...,NF , k∈U ∈ R(M+K)NF+that the cells will not be able to use or store.Our objective is to maximize the sum of the energy efficiencies of the two-tier cellswhile satisfying an average quality of service for each cell:(P )maximizep,pg ,ph,w∑c∈M∪K EEc (p,pg)subject to c1 : 1TT∑t=1∑u∈Uclog(1 + γu,t)≥ ρc, ∀c ∈M∪Kc2 :f∑i=1L∑l=1∑u∈Ucphu,(i−1)L+lTs ≤f−1∑i=0(Hc,i − ωc,i) , ∀c ∈M∪K, ∀f = 1, . . . , NF ,c3 :f∑i=1(Hc,i −L∑l=1∑u∈Ucphu,(i−1)L+lTs − ωc,i)≤ Bc, ∀c ∈M∪K, ∀f = 1, . . . , NF − 1.c4 :∑u∈Ucpu,t ≤ Pc, ∀c ∈M∪K, ∀t = 1, . . . , T,c5 : pgu,t + phu,t = pu,t, ∀u ∈ Uc, ∀c ∈M∪K, ∀t = 1, . . . , T,c6 : p ≥ 0, pg ≥ 0, ph ≥ 0, w ≥ 0.The first set of constraints c1 in (P ) consists of the average rate constraints for themacrocell and small cells. Next, the causal energy constraint c2 ensures that the totalrenewable power used by each BS for transmission up until the n-th frame will not exceed543.4. Energy Allocation Problemthe available amount that was harvested and stored in the previous f − 1 frames. Onthe other hand, constraint c3 means that the amount of renewable energy that each BScan store at the end of each energy-harvesting frame shall not exceed its battery capacitylimit. Consequently, the excess energy must be discarded through the auxiliary variablew. Then, the transmit power constraints for the BSs are given in c4. Finally, we have thepower usage equality constraints c5 introduced in (3.2) and the non-negativity constraintsc6 of the power vectors.This offline energy allocation problem described by (P ) is hard to solve due to itsnon-convexity. However, we will show that it is still possible to leverage the tools ofconvex optimization to find an efficient solution. In particular, we will use sequential andparametric convex programming techniques to derive a convergent algorithm for problem(P ).3.4.1 Minorization-Maximization with Non-convex Rate ConstraintsThere are intricate difficulties in solving problem (P ). First, the feasibility set implied bythe rate constraints c1−2 is not convex. In fact, the rate functions are not concave due tothe interference between the cells. Thereby, the objective function is not concave eitherbecause it is defined as a sum of ratios between the rate functions and the affine powercosts.To solve problem (P ), we use an extended version of minorization-maximization proce-dure, which also minorize the non-convex constraints. Given these additional complexities,553.4. Energy Allocation Problemwe will design a new algorithm that relies on the following convex subproblem (Qq):(Qq)maximizep,pg ,ph,w∑c∈M∪K E˜Ec (p,pg;q)subject to c1 : R˜c (p;q) ≥ ρcT, ∀c ∈M∪KConstraints c2−6,(3.6)where the vector q ∈ RNT+ is not an optimization variable but a parameter that defines theobjective function of (Qq).In fact, this new problem (Qq) is obtained from (P ) by replacing the non-concaveobjective function and the non-concave rate functions in c1−2 by their respective concaveminorizing functions. These new functions are respectively denoted by R˜c and E˜Ec for therate and energy efficiency of the cell c. For completeness, let us first recall the definitionof a minorization.Before we can solve (Qq), we need to obtain the minorizations for the rate and energyefficiency functions of each cell. For that, let us first define the following function r˜u for anarbitrary macro-cell or small-cell user u ∈ U :r˜u (p;q) ,T∑t=1logσ2u,t +∑j ∈ Uguj,tpj,t− li (p−u,q−u) (3.7)where p−u ∈ R(N−1)T+ is the concatenation of the power of all users except user u. Theaffine function lu in (3.7) is given bylu (p−u;q−u) =T∑t=1(vu,t (q−u,t) +∇vu,t (q−u,t)> (p−u,t − q−u,t)), ∀u ∈ U (3.8)563.4. Energy Allocation Problemwhere we havevu,t (p−u,t) = logσ2u,t +∑j∈U\{u}guj,tpj,t , ∀u ∈ U , ∀t (3.9)and∇vu,t (q−u,t) =guj,tσ2u,t +∑k∈U\{u}gukpk,tj∈U\{u}, ∀u ∈ U , ∀t. (3.10)Having defined the function r˜u for all u ∈ U , the following lemma gives the expressionsfor the minorizations of each cell’s rate and energy efficiency functions.Lemma 3.1. The minorizations of the rate functions {EEc}c∈M∪K and of the energy efficiencyfunctions {Rc}c∈M∪K of the cells are respectively given byE˜Ec (p,pgc ;q) =∑u∈Ucr˜u (p;q)Cc (pgc), ∀c ∈M∪K, (3.11)R˜k (p;q) =∑u∈Ucr˜u (p;q) , ∀c ∈M∪K. (3.12)Proof. The proof is similar to that of Lemma 2.2 in Chapter 2.After we found the minorizations, we now derive an algorithm for solving the originalproblem (P ) based on a similar procedure presented in Section 2.6. First we initialize theiteration index to n = 0 and start with a feasible power allocation p(0). By setting the pa-rameter q to p(n), we compute the minorizations of the energy efficiency and rate functionsusing equations (3.11)-(3.12). Then, we solve at each iteration n the surrogate problem(Qq) whose solution gives us a new power allocation p(n+1) as well as the correspondingpower usage vectors{pg (n+1),ph (n+1),w(n+1)}. The new power allocation p(n+1) is thenused to provide new minorizations that will be used for the next iteration. These steps arerepeated until convergence. A reasonable stopping criterion is when the increase of the573.4. Energy Allocation ProblemAlgorithm 3.1: Offline energy allocation algorithm for solving (P )Initialize: , p(0),pg (0),ph (0) ∈ RNT and w(0) ∈ R(M+K)NF . Set n = 0.Repeat(S.1) : Set q← p(n).(S.2) : Determine the minorizations R˜c, and E˜Ec, ∀c ∈M∪K using(3.11)-(3.12).(S.3) : Update the power allocation p(n+1) by solving problem (Qq) in (3.6).(S.4) : n← n+ 1Until (3.13) is satisfiedsum energy efficiency is less than some threshold :∑c∈M∪KEE(n+1)c −∑c∈M∪KEE(n)c ≤ . (3.13)This algorithm is summarized in Algorithm 3.1 and its convergence property is summarizedbelow.Proposition 3.1. Suppose we start Algorithm 3.1 with feasible power allocation p(0) andpower usage vectors{pg (0),ph (0),w(0)}. Then, all next iterates p(n) and{pg (n),ph (n),w(n)}produced in step (S.3) are also feasible ∀n ≥ 1. In addition, the algorithm monotonicallyconverges to at least a local optimum of the sum energy efficiency maximization problem (P ).Proof. See Appendix B.It is therefore important to start with initial power allocation and power usage vectorsthat satisfy all the constraints of (P ). For this purpose, we can find a feasible point by583.4. Energy Allocation Problemsolving first the following feasibility problem:(F )maximizep,pg ,ph,w,z∑c∈M∪K zcsubject to r1 : 1TT∑t=1∑u∈Uclog(1 + γu,t)+ zc ≥ ρc, ∀c ∈M∪KConstraints c2−6,z ≥ 0.For the feasibility problem (F ), we introduced new positive variable z , (zc)c∈M∪K ∈RM+K+ . Each slack variable zc corresponds to the violation of the average rate constraint ofcell c. Furthermore, we replaced the average constraints of (P ) by the relaxed constraintsr1. In contrast to (P ), the feasibility problem (F ) always admits a solution. When a solutionof (F ) satisfies∑c∈M∪K zc = 0, then it must be a feasible solution for (P ). Although (F ) isalso not convex, we can again apply the extended minorization-majorization procedure tofind a feasible solution for (P ). This procedure is listed in Algorithm 3.2.The proposed Algorithm 3.1 for maximizing the sum energy efficiency is therefore anascent algorithm. In particular, it offers the benefits of producing iterates that are alwaysfeasible. In addition, it is monotonically convergent. However, the step (S.2) in Algorithm3.1 depends on the solution of the surrogate problem (Qq). Although problem (Qq) isless complex than (P ), (Qq) is still not convex due to its fractional objective function.Therefore, we will next derive another algorithm that finds an optimal solution for problem(Qq).3.4.2 Parametric Convex OptimizationIn this subsection, we apply the convex parametric approach introduced in Section 2.5 tosolve the surrogate problem (Qq). By putting (Qq) in its epigraph form, we obtain the593.4. Energy Allocation ProblemAlgorithm 3.2: Solving the feasibility problem (F )Initialize: , p(0),pg (0),ph (0) ∈ RNT and w(0) ∈ R(M+K)NF . Set n = 0.Repeat(S.1) : Set q← p(n).(S.2) : Determine the minorization R˜c, ∀c ∈M∪K using (3.12).(S.3) : Solve the convex optimization problem:maximizep,pg ,ph,w,z∑c∈M∪K zcsubject to R˜c (p;q) + zc ≥ ρcT, ∀c ∈M∪KConstraints c2−6,z ≥ 0.(S.4) : n← n+ 1Until∑c∈M∪K z(n)c ≤ Output: Feasible power allocation and usage vectors{p,pg,ph,w}.equivalent reformulation (Eq):(Eq)maximizep,pg ,ph,w,θ∑c∈M∪K θcsubject to c0 : R˜c (p;q)− θcCc (pgc) ≥ 0, ∀c ∈M∪K,(p,pg,ph,w)∈ S,(3.14)where the auxiliary variable θ , (θ1, . . . , θM+K) and the new constraints c0 are introducedto replace the fractional objective problem by a linear term. For convenience, we alsoreplaced the convex constraints c1−6 of (Qq) into the convex feasibility set S.With these manipulations, we have maintained the equivalence between (Eq) and (Qq).Nonetheless, the two formulations are different. While (Qq) has a non-convex objectivefunction and convex constraints, the equivalent problem (Eq) has a linear objective func-tion, a new variable θ and new non-convex constraints c0. Next, we will employ the convexparametric optimization approach to construct an algorithm that finds an optimal solution603.4. Energy Allocation Problemfor (Eq), and hence for (Qq).Let us consider the following family of parametric convex problems (Px):(Px)maximizep,pg ,ph,w∑c∈M∪K αc(R˜c (p;q)− θcCc (pgc))subject to(p,pg,ph,w)∈ S(3.15)which is parameterized by the vector of parameters x , (α, θ) with α , (α1, . . . , αM+K).Observe that the vector θ is not an optimization in (Px) but a parameter. This parametricproblem can be interpreted as a weighted sum utility maximization in which each cell’sutility is the difference of its sum rate and a pricing penalty term. More importantly, (Px)is a convex problem and can be used as a vehicle to solve (Eq) due to the following result.By applying the result in Proposition 2.1, if (Eq) is feasible and admits an optimalsolution{p̂, p̂g, p̂h, ŵ, θ̂}. Then, there exists α̂ such that{p̂, p̂g, p̂h, ŵ}is a solution of(Px) with x =(α̂, θ̂). Furthermore, the solution{p̂, p̂g, p̂h, ŵ}must satisfy the followingsystem of nonlinear equations:F (x;q) = 0, (3.16)where the vector function F : R2(K+M) → R2(K+M) is defined by its component functionsas follows:Fc (x;q) = θcCc (p˜gc (x))− R˜c (p˜ (x) ;q) , ∀c ∈M∪K, (3.17)Fc+M+K (x) = αcCc (p˜gc (x))− 1, ∀c ∈M∪K. (3.18)These component functions are obtained by composing the minorizations{R˜c}c∈M∪Korthe power costs {Cc}c∈M∪K with the functions p˜ (x) and {p˜gc (x)}c∈M∪K that return anoptimal power allocation of the parametric problem (Px) for a given parameter x.Furthermore, the uniqueness of the root of the nonlinear system F (x;q) = 0 wasestablished in Proposition 2.2. These previous results lead us to a simpler solution method613.4. Energy Allocation Problemfor the non-convex problem (Qq). In fact, whenever (Qq) is feasible, an optimal solution{p̂, p̂g, p̂h, ŵ}can be obtained by first solving the system F (x;q) = 0 and then by solvingthe parametric convex problem (Px) with x set to the unique root of the system. In otherwords, solving the non-convex problem (Qq) reduces to solving a nonlinear system ofequations. In the next subsection, we present an iterative algorithm based on Newtonmethod which enables us to find the root of F (x;q) = 0 and solve (Qq).3.4.3 Damped Newton Algorithm for Solving (Qq)Our proposed algorithm for finding the optimal parameter x and solving (Qq) is based ona damped Newton method [63]. First, we start with the initial power allocation vector p(0)of all users and the initial power usage vectors{pg (0),ph (0),w(0)}of all cells. Then, weinitialize the parameter x(0) =(α(0), θ(0))as follows:α(0)c =1Cc(p˜g (0)c) ,∀c ∈M∪K, (3.19)θ(0)c =R˜c(p(0))Cc(pg (0)c) , ∀c ∈M∪K. (3.20)At each iteration n ≥ 0 of the algorithm, we solve the convex parametric program (Px)that is parameterized by x (n). This gives us a new power allocation vector p˜(x(n))andpower usage vectors{p˜g(x(n)), p˜h(x(n)), w˜(x(n))}. After that, we check if the stoppingcriterion∥∥F(x(n))∥∥ <  is satisfied for a given tolerance  > 0. If it is satisfied, then we stopthe algorithm. Otherwise, we update the parameter x(n) using a damped Newton method.To do so, we first compute the Newton step at the n-th iteration as follows:d(n) = −F′(x(n))−1F(x(n)), (3.21)where F′ is the Jacobian matrix of the vector-valued function F. In fact, it can be shownthat F is differentiable and that its Jacobian F′ is a non-singular, diagonal and positive623.4. Energy Allocation Problemdefinite matrix given by [81]:F′ (x) =diag (c) 0(M+K)×(M+K)0(M+K)×(M+K) diag (c). (3.22)where the vector c concatenates the power costs of all cells as follows:c =(Cc(p˜g(x(n))))>c∈M∪K∈ RM+K+ .Since the matrix F′ (x) is diagonal and positive definite, its inverse can be easily cal-culated. Once we get the Newton step from (3.21), we perform a line search to find aneffective step size 0 < λ < 1. Precisely, we start with the full Newton step, i.e. λ = 1and reduce it by a constant factor 0 < σ < 1 until a sufficient decrease in the l2-norm ofF(x(n) + λd(n))is obtained. After that, we update the parameter vector as:x(n+1) , x(n) + λd(n). (3.23)or component-wise as follows:α(n+1)c = (1− λ)α(n)c + λ1Cc (p˜g (x(n))), (3.24)θ(n+1)c = (1− λ) θ(n)M + λR˜c(p˜(x(n));q)Cc (p˜g (x(n))), (3.25)Then, we use this new parameter x(n+1) for the next iteration and we repeat this iter-ative procedure until convergence. This proposed iterative damped Newton algorithm issummarized in Algorithm 3.3. Its convergence is stated in the following proposition.According to Proposition 2.3, Algorithm 3.3 converges to an optimal solution of theconvex subproblem (Qq) in (3.6).633.5. Joint Energy Allocation and Energy CooperationAlgorithm 3.3: Subprocedure for solving the surrogate problem (Qq).Input: Parameter q and constants , 0 < (µ, λ, σ) < 1.Initialize: p(0) ∈ RNT and set n = 0Repeat(S.1) : Solve problem (Px) to obtain p˜(x(n)), p˜g(x(n)), p˜h(x(n))and w˜(x(n)).(S.2) : Initialize λ← 1,(S.3) : Compute x(n) + λd(n) using (3.24)-(3.25).,(S.4) : while∥∥F(x(n) + λd(n))∥∥ > (1− µλ)∥∥F(x(n))∥∥ do λ← σλ(S.5) : Update x(n+1) ← x(n) + λd(n)(S.6) : n← n+ 1Until∥∥F(x(n))∥∥ ≤ Result: Root x(n) of (3.16).3.5 Joint Energy Allocation and Energy CooperationIn this section, we propose an energy allocation and cooperation scheme for the wirelesstwo-tier network. Again, we assume that the cells are powered by hybrid power sources. Inaddition, the cells can exchange their harvested energy through a smart-grid infrastructure.In fact, recent advancements in smart-grid will integrate the distributed generation ofrenewable energy into conventional power grid. This will be achieved by enabling a two-way energy flow between distributed energy micro-grid generators and the smart-grid[41]. In this work, we explore this future possibility to efficiently power heterogeneousnetworks. Particularly, if a cell harvests more renewable energy than it can store at aspecific frame, then it can inject a portion of that excess energy to the smart-grid insteadof wasting it. On the other hand, a cell that could not harvest or store enough energycan benefit by drawing some of the transferred renewable energy. As illustrated in Figure3.2, we assume that there exists a micro-grid aggregator that controls the energy sharingbetween the base stations. Thus, the aggregator helps realize the two-way energy flowbetween the cells using real-time information about their energy usage. Here, we assumethat the energy transfer happens on a per-frame basis. At the end of each frame, if there isstill an excess energy that cannot be stored by all cells, that energy will be diverted by theaggregator into the smart-grid for other uses.643.5. Joint Energy Allocation and Energy CooperationHybrid PowerSupplySolar PanelWind TurbineSmartmeterSmartmeterEnergy owData owSmartmeterMicro-gridAggregatorSmart GridFigure 3.2 – A model of cellular networks with hybrid power source and energy sharing through a smartmicro-grid.3.5.1 Energy Cooperation between Two-tier CellsTo model the energy cooperation, we introduce the following array of energy exchangevariables−→H =(−→h c,d,i)c,d∈M∪K,i∈1,...,NF∈ R(M+K)×(M+K)×NF+ . Each element−→h c,d,i ≥ 0denotes the amount of harvested renewable energy that a cell c transfers to another celld at the end of frame i. When cell c transfers an energy−→h c,d,i through the smart-gridpower network, we assume that there is an energy loss that is incurred by the transfer. Asa result, the effective energy that cell d can receive is η−→h c,d,i with 0 < η < 1 being thetransfer efficiency factor.Next, we define by the auxiliary variable δc,i as the net energy injected by a cell c intothe aggregator at the end of frame i as:δc,i =∑d∈M∪K\{c}(−→h c,d,i − η−→h d,c,i), ∀c ∈M∪K, ∀i = 0, ..., NF − 1. (3.26)653.5. Joint Energy Allocation and Energy CooperationIf δc,i is positive, then the cell c is effectively injecting some renewable energy into thesmart-grid. Conversely, it is drawing renewable energy from the smart-grid when δc,i isnegative. Note that our model does not explicitly preclude a cell from drawing and inject-ing some renewable energy from/into the grid at the same time. Since our focus here is onoffline optimization, we need not explicitly add this constraint in our problem formulation.After defining these auxiliary variables, we modify the energy causality constraints forthe energy-cooperative two-tier cells as follows:f∑i=1L∑l=1∑u∈Ucphu,(i−1)L+lTs ≤f−1∑i=0(Hc,i − ωc,i − δc,i) , ∀c ∈M∪K, ∀f = 1, . . . , NF . (3.27)Similarly, the battery constraints at the cells become:f∑i=1(Hc,i −L∑l=1∑u∈Ucphu,(i−1)L+lTs − ωc,i − δc,i)≤ Bc, ∀c ∈M∪K, ∀f = 1, . . . , NF − 1(3.28)For the energy cooperation model, we use the following network energy efficiencymetric:EE0 (p,pg) ,∑c∈M∪KRc (p)C0 (pg)where the total network power consumption C0 is equal to:C0 (pg) =∑c∈M∪K[P fc + ψT∑t=1∑u∈Ucpgu,t]. (3.29)Because of the energy cooperation between the cells, we use this network energy efficiencymetric as a new objective function. By cooperating, the cells can exchange their harvestedrenewable energy and share their battery storage.663.5. Joint Energy Allocation and Energy CooperationAs a result, the problem formulation for the energy allocation and cooperation is givenbelow:(C)variables p,pg,ph,w,−→H, δmaximize EE0 (p,pg)subject to d1 : 1TT∑t=1∑u∈Uclog(1 + γu,t)≥ ρc, ∀c ∈M∪K,d2 :f∑i=1L∑l=1∑u∈Ucphu,(i−1)L+lTs ≤f−1∑i=0(Hc,i − ωc,i − δc,i) , ∀c ∈M∪K, ∀f = 1, . . . , NF ,d3 :f∑i=1(Hc,i −L∑l=1∑u∈Ucphu,(i−1)L+lTs − ωc,i − δc,i)≤ Bc, ∀c ∈M∪K, ∀f = 1, . . . , NF − 1,d4 :∑u∈Ucpu,t ≤ Pc, ∀c ∈M∪K, ∀t = 1, . . . , T,d5 : pgu,t + phu,t = pu,t, ∀u ∈ Uc, ∀c ∈M∪K, ∀t = 1, . . . , T,d6 :∑d∈M∪K\{c}(−→h c,d,i − η−→h d,c,i)= δc,i, ∀c ∈M∪K, ∀i = 0, ..., NF − 1,d7 : p ≥ 0, pg ≥ 0, ph ≥ 0, w ≥ 0,−→H ≥ 0.In next subsection, we present an algorithm for solving this new problem.3.5.2 Offline Optimization AlgorithmSimilar to (P ), the problem (C) above is also non-convex. However, it is more tractableas its objective function consists of one fractional term. Following the approach in theprevious section, we handle the non-convexities in (C) using the extended minorization-majorization procedure. After that, we use the simple Dinkelbach method [9] to solve theminorized subproblem.673.5. Joint Energy Allocation and Energy CooperationFirst, let us write the expression for the minorization E˜E0 of the objective function of(C) as:E˜E0 (p,pg;q) =∑c∈M∪K R˜c (p)C0 (pg). (3.30)Next, we consider the following minorized subproblem associated to (C):(Mq)variables p,pg,ph,w,−→H, δ, θmaximize θsubject to∑c∈M∪K R˜c (p;q)− θC0 (pg) ≥ 0,R˜c (p;q) ≥ ρcT, ∀c ∈M∪K,Constraints d2−7,(3.31)which is obtained by first minorizing the rate functions in (C) at q, and then by rewritingthe resulting problem in its epigraph form. This problem (Mq) is analogous to problem(Qq) in (3.6). To solve (Mq), we use the following convex parametric subproblem, whichis parameterized by both θ and q:(Dθ,q)variables p,pg,ph,w,−→H, δmaximize∑c∈M∪K R˜c (p;q)− θC0 (pg)subject to R˜c (p;q) ≥ ρcT, ∀c ∈M∪K,Constraints d2−7.(3.32)After formulating the subproblem (Dθ,q), we now propose the Algorithm 3.4 for solving683.5. Joint Energy Allocation and Energy CooperationAlgorithm 3.4: Algorithm for solving the energy management and cooperation prob-lem (C)Input: Constants  and δInitialize: p(0),pg (0),ph (0) ∈ RNT and w(0) ∈ R(M+K)NF .−→H(0) ∈ R(M+K)×(M+K)×NF+ with feasible values for (C). Set n = 0, k = 0.Repeat(S.1) : Set the parameter q(n) ← p(k).(S.2) : Compute the minorizations{R˜c}c∈M∪Kand E˜E0 at q(n) using (3.12)and (3.30).(S.3) : Initialize θ(k) =∑c∈M∪K R˜c(p(k);q(n))/C0(pg (k)).(S.4) : while∥∥∥∑c∈M∪K R˜c(p(k);q(n))− θ(k)C0(pg (k))∥∥∥ >  doSolve the convex problem(Dλ(k),q(n))to obtain{p̂, p̂g, ŵ,−̂→H},Update λ(k+1) ← λ(k) −∑c∈M∪K R˜c(p̂;q(n))C0(p̂c),Update p(k+1) ← p̂,k ← k + 1(S.5) : n← n+ 1Until EE(n+1)0 −EE(n)0EE(n)0≤ δOutput: Power allocation p and power usage vectors{p,pg,ph,w,−→H}.(C), which is listed in the next page. It consists of two loops. In each iteration k of theinner loop, we solve a convex subproblem(Dλ(k),q(n))that is parameterized by both λ(k) andq(n). At the end of that iteration, we update the parameter λ(k) using a full Newton step.In fact, a linear search is not required in the Dinkelbach method to guarantee theoreticalconvergence to an optimal solution of (Mq). We exit the inner loop when the value of theobjective function of(Dλ(k),q(n))becomes zero. After that, we update the parameter q(n) ateach iteration n of the outer loop and compute new minorizations of the rate and objectivefunctions using the new parameter. Then, the process is repeated until the increase in thenetwork energy efficiency is smaller than a specified tolerance δ. Similar to Algorithm 3.1,this new algorithm enjoys the same convergence properties as those stated in Proposition2.4.693.6. Simulation Results−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1−0.8−0.6−0.4−0.200.20.40.60.81MBSSC 1SC 2SC 3SC 4Figure 3.3 – Base stations and user locations.3.6 Simulation ResultsNext, we present our simulation results for a wireless two-tier networks with M = 1macrocell and K = 4 small cells. The MBS serves 3 macrocell users (MUs) whereas theSBSs each serve one small-cell user (SU). The SBSs and all users are randomly placed in a1km by 1km grid region as shown in Figure 3.3. The MUs and SUs are denoted by trianglesand rectangles respectively.We model the large-scale path loss attenuation PL (in dB) using the WINNER model[74] described in (2.55). The simulation parameters are listed in Table 3.2. In addition, wemodel the time-varying small-scale fading as a correlated Rayleigh process in accordancewith Clarke’s scattering model [69]. This was simulated using the autoregressive modelingtechnique in [80]. Taking these into account, the time variations of the channel gains forthe first and second SUs are, for instance, illustrated in Figure 3.4.703.6. Simulation ResultsTable 3.2 – Path loss model and system parameters in Chapter 3.Parameters ValuesCarrier frequency fc = 1.9 GHzSBS-to-SU path loss (model A1-LOS) A = 1.87, B = 46.8, C = 20SBS-to-MU path loss (model A1-NLOS) A = 3.68, B = 43.8, C = 20MBS-to-SU path loss (model C1-NLOS) A = 3.36, B = 44.36, C = 23MBS-to-SU path loss (model C1-NLOS) A = 3.36, B = 44.36, C = 23Standard deviation of SBS-to-SU shadowing S = 3 dBStandard deviation of SBS-to-PU shadowing S = 4 dBStandard deviation of MBS-to-SU shadowing S = 6 dBMBS antenna power gain 12dBSBS antenna power gain 5dBNormalized maximum Doppler frequency fdTS = 0.01Total number of frames NF = 10Time slot duration Ts = 1secLength of energy-harvesting frame L = 10 time slots713.6. Simulation Results0 10 20 30 40 50 60 70 80 90 100−40−30−20−1001020TimeslotChannel gain (dB)  MBSSBS 1SBS 2SBS 3SBS 4(a) Channels of SU #10 10 20 30 40 50 60 70 80 90 100−50−40−30−20−1001020TimeslotChannel gain (dB)  MBSSBS 1SBS 2SBS 3SBS 4(b) Channels of SU #2Figure 3.4 – Channel gains for the desired signal (solid line) and inter-cell interference (dotted line) receivedby SUs.3.6.1 Convergence of AlgorithmsFirst, we present the convergence of the proposed Algorithm 3.1. These algorithms wereimplemented in Matlab and we used the disciplined convex optimization software CVXto solve the convex parametric subproblems [1]. In Figure 3.5(a), we observed a fastconvergence as the norm of F (x) , which is defined in (3.17)-(3.18), rapidly decreasedtowards zero. Figure 3.5(b) shows that the energy efficiency of each cell also convergesaccordingly. For this example, only one inner iteration of the damped newton algorithmwas needed to solve each subproblem (Qq). In other words, a full Newton step satisfiedthe norm-decrease condition. However, the line search is in general required to guaranteeconvergence. In our experiments, the algorithm converges within 10 to 20 inner iterationsin most cases.3.6.2 Analysis of Energy Efficiency PerformanceNext, we analyze the energy efficiency of the proposed schemes. In our simulation, wenormalized the transmit power Pc, the battery capacityBmax and the energy-harvesting rate723.6. Simulation Results1 2 3 4 5 610−410−310−210−1100101102103IterationFx ()(a) Value of the norm∥∥F(x(n))∥∥1 2 3 4 5 60102030405060IterationEnergy efficiency (bps/J)  MacrocellSC 1SC 2SC 3SC 4Sum EE(b) Value of minorized energy efficiency per cellFigure 3.5 – Convergence of Algorithm 3.1.of the renewable source by the noise power σ2. Without loss of generality, we assume thatthe time slot duration Ts = 1sec to simplify the analysis. We set the maximum normalizedtransmit power of the SBS and MBS to 10 dB and 13 dB respectively. Then, the normalizedbattery capacity Bmax is set to be equal to 10 dB and 17 dB. The main reasons to normalizethe energy arrival rate and battery capacity to a decibel scale are to simplify the analysisand to facilitate reproducibility of our simulations. Since the contributions of this chapteris mainly theoretical, future studies could use the results presented here for performancebenchmark. In such case, normalized energy levels instead of absolute energy in Wattshelps abstract away implementation-specific details such as the type of energy storagesystem or energy sources.In our simulations, we consider three different models for the renewable energy arrivalper frame: (1) constant rate, (2) linearly increasing rate and (3) Poisson random variable.Although the energy arrival is commonly modeled as a Poisson renewal process [10, 82],the constant and linear rate models help gain some insight on the energy allocation. Forfair comparison, we set the average harvested energy to be equal for these three scenarios.In addition, the energy rate is assumed to be equal for each cell. For the Poisson model,733.6. Simulation Results0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 15060708090100110120130140150Normalized harvesting rates (H/Bmax))Sum Energy Efficiency (bps/J)  Constant rateLinear increasingPoisson arrivalBmax = 17 dBBmax = 10 dBBmax = 10 dBEnergy Coop.Figure 3.6 – Comparison of sum energy efficiency for different energy harvesting rates and arrivals.we average the achieved energy efficiency over 100 different realizations of the energyarrivals.Figure 3.6 shows the achieved sum energy efficiency for different energy harvestingrates. As expected, the energy efficiency performance increases with the energy arrivalrate as this later is increased from 0.1Bmax to Bmax per frame. Moreover, we see thatincreasing the capacity of the energy storage at each cell to Bmax = 17 dB significantlyimproves the energy efficiency. Finally, Figure 3.6 shows that under the same capacitylimit Bmax = 10 dB, the energy-efficiency performance can be improved by exploiting theenergy cooperation between the cells.In the next Figure 3.7, we look at the energy efficiency at each cell when arrival rateis increased from 0.1Bmax to Bmax per frame. We notice similar patterns for the per-cellenergy efficiency under the three different assumptions for the energy arrival rates. SinceSU#1 has favorable channel conditions (see Figure 3.4), the energy efficiency gain of743.6. Simulation Results1 2 3 4 50510152025303540Cell index & Normalized harvesting rateEnergy efficiency (a) Constant arrival rate1 2 3 4 50510152025303540Energy efficiency Cell index & Normalized harvesting rate(b) Linearly increasing rate1 2 3 4 50510152025303540Cell index & Normalized harvesting rateEnergy efficiency (c) Poisson arrivalFigure 3.7 – Energy efficiency per cell for energy-harvesting rates between 0 dB and 10 dB.small-cell (SC) #1 (cell index 2 in Figure 3.7) is very high when the arrival rate increasesfrom 0.1Bmax to Bmax. In contrast, small-cell #2 (cell index 3) suffers from strong inter-cell interference and its energy efficiency remain the same. Finally, Figure 3.8 shows thegrid power consumption of each cell over time. It is evident from Figure 3.8 that energycooperation decreases the required energy drawn from the conventional grid source.753.6. Simulation Results0 10 20 30 40 50 60 70 80 90 100012345678910TimeslotGrid power consumption  MacrocellSC 1SC 2SC 3SC 4(b) No Cooperation - RateEH = 0dB0 10 20 30 40 50 60 70 80 90 100012345678910TimeslotGrid power consumption  MacrocellSC 1SC 2SC 3SC 4(c) No Cooperation - RateEH = 6dB0 10 20 30 40 50 60 70 80 90 100012345678910TimeslotGrid power consumption  MacrocellSC 1SC 2SC 3SC 4(d) No Cooperation - RateEH = 10dB0 10 20 30 40 50 60 70 80 90 100012345678910TimeslotGrid power consumption  MacrocellSC 1SC 2SC 3SC 4(f) Energy Cooperation - RateEH = 0dB0 10 20 30 40 50 60 70 80 90 100012345678910TimeslotGrid power consumption  MacrocellSC 1SC 2SC 3SC 4(g) Energy Cooperation - RateEH = 6dB0 10 20 30 40 50 60 70 80 90 100012345678910TimeslotGrid power consumption  MacrocellSC 1SC 2SC 3SC 4(h) Energy Cooperation - RateEH = 10dBFigure 3.8 – Grid power consumed per cell for different Poisson arrival rates.76Chapter 4Power minimization of CooperativeClustered Small-cell Networks4.1 IntroductionIn Chapter 2 and Chapter 3, we consider two-tier networks in which the base stationshave single antennas. In this chapter, we assume that the base stations can have multipleantennas. Therefore, we extend the power allocation to a cooperative multi-cell beam-forming scheme, which is also known as coordinated multi-point transmission (CoMP)[83] or network MIMO. By enabling cooperation between the two-tier cells, we can reaplarger capacity and diversity gains [84] and also to mitigate the interference [85]. This isachieved thanks to the increased spatial degrees of freedom of CoMP systems.Despite these benefits of CoMP, it can also come with significant costs [11, 86]. First,the cooperating BSs must be connected by a backhaul through which they exchange thechannel knowledge and user data. Second, cooperation largely increase the energy con-sumption due to the extra signal processing [11]. Although the theoretical benefits andpractical issues of CoMP have been studied in many works, only few have looked into itsenergy efficient implementation.In this chapter, we thus propose a flexible cooperation scheme which balances thetrade-off between performance and energy efficiency. Instead of maximizing performanceas in previous works [87–89], our goal is to enable the cells to satisfy their users andprotect the primary users while minimizing their overall power consumption.774.1. IntroductionIn Section 4.6, we present the state-of-the-art research on green CoMP and distributedantenna systems. Then, in Section 4.3, we propose a flexible and energy-aware coopera-tion scheme for cellular two-tier systems. Using the energy consumption model in [11],we formulate a cross-layer optimization of cooperative beamforming and clustering as amixed integer program. By jointly optimizing the clustering and beamforming, we providean extra degree of freedom for the cells to adapt their transmission according to changingparameters such as the users’ locations, service requirements and channel conditions. Wealso propose a cognitive resource allocation mechanism to enable the spectrum sharingbetween the macrocell and small cells.In Section 4.4, we present a framework for decomposing this problem into a clusteringmaster problem and a beamforming subproblem. Then, we derive an optimally conver-gent algorithm using the generalized Benders decomposition (GBD) method [90]. We alsopropose simple techniques for accelerating the convergence of this algorithm.Then, we propose a decentralized optimization algorithm for the cooperative beam-forming when the clustering is fixed in Section 4.5. Assuming that the clusters do notoverlap, we exploit the quasi-separability of the problem and derive a distributed algo-rithm based on the alternating direction method of multipliers (ADMM) [91]. We alsopropose limited signaling schemes that can considerably reduce the overhead.Finally, numerical simulations are presented in Section 4.6 to analyze the trade-off be-tween performance and energy consumption in a cooperative two-tier systems. In addition,we examine the convergence and effectiveness of the proposed algorithms. In Section 4.6,we also study the performance of the proposed decentralized beamforming and signalingschemes.784.2. Related Works4.2 Related WorksThe study of multi-cell joint-transmission system started with the landmark paper in [84].Within the decade that followed [84], many research efforts followed on looking at thetheoretical benefits of CoMP system from the viewpoint of spectrum efficiency [83]. Sev-eral papers have also been published on the practical aspects and issues, such as the impactof limited backhaul signaling [36] or the need for antenna selection and clustering [92].However, the energy efficiency of CoMP has not received much attention until later.For instance, [11] was the first to look at the energy consumption model of cooperatingbase stations. In [93], an energy efficiency analysis of an idealized CoMP system was pre-sented. The trade-off between energy efficiency and spectrum efficiency has been studiedby [94]. Then, [45] studied the energy-efficient power allocation in distributed antennasystems. However, [45] assumed a zero-forcing strategy and did not try to optimize thecooperative beamforming. However, zero-forcing beamforming is limited by the numberof antennas available at each base station. Therefore, a small number of users can beaccomodated to avoid interference between the multi-cell transmissions. [95] tackled thedifficult problem of energy-efficient transmit precoding design using a parameterizationof the Pareto boundary. They proposed an iterative method in which each user optimallysolves a parameterized energy-efficiency maximization problem and updates the inter-userinterference parameters at each iteration. Nonetheless, they could not prove the optimalconvergence of such algorithm. In contrast, we propose in this chapter a multi-cell beam-forming optimization scheme for small-cell networks. Instead of solving an energy effi-ciency maximization problem, we aim to minimize the overall energy consumption of theCoMP system. In contrast to previous works, we propose a scheme that jointly optimizesthe clustering and the beamforming.After the results of this chapter were published in [96], there were other papers thatconsidered the energy efficiency of CoMP. For instance, [97] gave a performance analysisand designed an optimization scheme for green heterogeneous CoMP networks. Another794.3. System Modelstudy in [98], which is based on a similar idea proposed in this chapter, investigates dif-ferent base station sleeping strategies for energy-efficient small-cells.4.3 System ModelWe consider the downlink of a clustered cooperative small-cell networks whose base sta-tions B ={b1, . . . , b|B|}are connected through a backhaul interface. Each base station hasm antennas. By sharing their antennas, the small-cell base stations can cooperate to servea pool of users U ={u1, . . . , u|U|}. Here, we make the following assumptions. The smallcells also co-exist with a primary macro-cell system using a cognitive underlay strategy.In other words, they share the spectrum spectrum under the condition that the secondarysmall-cell transmissions do not exceed specific interference limits that are imposed by theprimary macrocell users.We assume perfect channel information and perfect synchronization for the coopera-tive beamforming. Next, the backhaul interface is assumed to have enough capacity forexchanging the user data and signaling information. Hence, we consider the followingcooperation schemes for the secondary small cells:1. Full cooperation is achieved when all base stations cooperate to jointly serve theusers U . Essentially, this strategy corresponds to a m |B| × |U| network-MIMO trans-mission [84]. It provides the highest spatial degrees of freedom for multiplexing andfor interference mitigation but incurs the highest cost in terms of signaling and en-ergy consumption.2. Inter-cell coordination is the simplest scheme as each user is served by a singlebase station. In this case, the interference is mitigated by having the base stationscoordinate the base station-to-user assignment and beamforming.804.3. System Model3. Flexible cooperation allows each user u to be served either by a single base stationor by multiple cooperating ones. With this flexible strategy, it is thus possible tobalance performance and energy saving.In our model, we select the flexible cooperation as a general strategy and consider fullcooperation and inter-cell coordination as special cases. Let us define a binary vectorx = (xbu)b∈B, u∈U ∈ {0, 1}|B|·|U|×1 to model the clustering, where |S| denotes the cardinal ofa set S. The component xbu of x determines whether base station b serves the user u ornot:xbu =1 if user u is served by BS b0 otherwise(4.1)Thus, the special case of full cooperation corresponds to x = 1|B|·|U|×1 so that eachuser is jointly served by all base stations. On the other hand, the special case of inter-cellcoordination requires the following equality to hold:∑b∈Bxbu = 1 ∀u ∈ U (4.2)to enforce that one and only one base station will be assigned to each user u.Next, let us describe the signal model by first assuming every base stations serve eachuser. Let hu , (hbu)b∈B ∈ Cm|B|×1 be the channel vector from all base stations to a useru where hbu ∈ Cm×1 is the channel between base station b and user u. Each channelvector hbu accounts for both the large-scale path loss attenuation as well as the small-scale Rayleigh fading. Similarly, vu , (vbu)b∈B ∈ Cm|B|×1 is the concatenated beamformingvector for user u with vbu ∈ Cm×1. Then, all base stations cooperatively transmit the datasymbol su of a user u through the channel hu (see Figure 4.1). Its received signal is thusgiven by:yu = hHu vusu +∑j∈U\{u}hHu vjsj + nu (4.3)814.3. System ModelTable 4.1 – List of key parameters in Chapter 4.Symbol Parameter descriptionB Set of base stationsU Set of users|B| Number of base stations|U| Total number of usersxbu Binary indicator for user u being served by BS byb Binary activity status of BS bhu Channel vector from all base stations to a user uhbu Channel vector between base station b and user u.vu Concatenated beamforming vector for user uvbu Beamforming vector of BS b for user uSINRu SINR achieved by user uρu Minimum SINR requirement for user uRu Achieved rate of user uPb Transmit power limit of BS bV Set of primary usersgbv Channel from the BS b to the primary user v.Ptotal Total power consumption of the systemPtx Transmit power radiated by all base stationsPsp Total power cost due to multi-cell processingPbh Backhaul power costP0 Total fixed power cost of the BSs824.4. Joint Clustering and Beamforming OptimizationJointly interferedPrimary UserJointly servedUserSignaling exchangeCore networkUser data User dataData exchangehu gvu vFigure 4.1 – Coordinated multi-point transmission model in distributed and cognitive small-cell networks.where the symbol (·)H means the conjugate transpose. Moreover, nu ∼ CN (0, σ2u) is acircularly complex Gaussian noise with zero-mean and a variance σ2u. Therefore, the signal-to-interference-plus-noise ratio (SINR) of the received signal yu is:SINRu =∣∣hHu vu∣∣2∑j∈U\{u} |hHu vj|2 + σ2u(4.4)and its achievable rate is given by:Ru = log (1 + SINRu) [bps/Hz] (4.5)4.4 Joint Clustering and Beamforming OptimizationOur objective is to minimize the total power consumption of the system given SINR targetsof the small-cell users and interference power constraints at the primary users.4.4.1 Beamforming Power ConstraintsEach small-cell base station is subject to power limit constraints:∑u∈U‖vbu‖22 ≤ Pb, ∀b ∈ B (4.6)834.4. Joint Clustering and Beamforming OptimizationTo model the flexible cooperation scheme, we make sure that if a specific base stationb does not serve user u , i.e. when xbu = 0, then the corresponding beamforming vectorvbu becomes a zero vector. In other words, we enforce that a base station b allocates anon-zero power for user u only if xbu = 1, through the mixed-integer constraints:‖vbu‖22 ≤ xbu · Pb, ∀u ∈ U , ∀b ∈ B (4.7)With constraints (4.6), the mixed integer constraint (4.7) simply becomes redundant whenxbu = 1. Although the SINR definition in (2.1) initially assumes full cooperation, it is alsovalid for the flexible cooperation scheme after we added constraints (4.7). In fact, when asubset of BSs B ⊂ B does not serve user u, then constraints (4.7) enforce ‖vbu‖2 = 0, ∀b ∈B.In addition, we assume that the two-tier network can co-exist with a primary systemwhose users are V ={v1, . . . , v|V|}. Using a cognitive underlay strategy, the small-cell basestations opportunistically use the same channel licensed to the primary system under thecondition that they do not exceed a total interference temperature limit Iv ≥ 0 imposed byeach primary user v as follows:∑b∈B∑u∈U∣∣gHbvvbu∣∣2 ≤ Iv, ∀v ∈ V (4.8)with gbv ∈ Cm×1 being the channel from the small-cell base station b to the primary user v.Note that these constraints couple the transmissions of the small-cell base stations.4.4.2 Energy Consumption ModelA linear model for the energy consumption of a CoMP system was given by [11, 58] asfollows:Ptotal = Ptx + Psp + Pbh + P0 (4.9)844.4. Joint Clustering and Beamforming Optimizationwhere Ptotal, Ptx, Psp, Pbh and P0 denote the total power consumption, the transmit powerradiated by all base stations, the signal processing power cost, the backhaul power costand the fixed power cost respectively. Except for the static consumed power, these costsscale dynamically with the load, i.e. the number of wireless links between the users andtheir serving base stations, as follows [11, 58]:Ptx =∑b∈B∑u∈U ‖vbu‖22µPA(1 + CPS)Psp =∑b∈B∑u∈UxbupspPbh =∑b∈B∑u∈Uxburupbhwhere ru is the data rate for user u, pbh is the power consumed for transmitting one bit ofinformation over the backhaul and psp is the power consumed per link for the cooperativeprocessing. Note that in contrast to the cooperation schemes studied in Chapter 2 andChapter 3, the multi-cell cooperation considered in this chapter may lead to significantenergy cost and this cost scales with the cluster size of the cooperating base stations [11].Finally, the static power P0 is the fixed power consumed by the active base stations:P0 =∑b∈Bybpstatic (4.10)where the binary variable yb defines the activity status of base station b such that:yb =1 if base station b is actively serving some user(s)0 otherwise(4.11)Therefore, we can define the overall power consumption of the network by a costfunction f of the beamforming variable v and of the binary variables x and y:f (x,y,v) =1 + CPSµPA·∑b∈B∑u∈U‖vbu‖22 +∑b∈B∑u∈Uxbu (psp + rupbh) +∑b∈Bybpstatic (4.12)= f1 (v) + f2 (x,y) (4.13)854.4. Joint Clustering and Beamforming Optimizationwhere the term f1 in (4.12) is the transmit-dependent power and the term f2 is the sum ofthe cooperative processing power and fixed power. Note that v , (vu)u∈U concatenates allbeamforming vectors of small-cell users.4.4.3 Problem FormulationFinally, we formulate the joint optimization of clustering and cooperative beamformingas:variables x y vminimize f (x,y,v) (4.14)subject to xbu ∈ {0, 1} , ∀u ∈ U , ∀b ∈ B (4.15)yb ∈ {0, 1} , ∀b ∈ B (4.16)yb ≤∑u∈Uxbu, ∀b ∈ B (4.17)yb ≥ xbu, ∀u ∈ U , ∀b ∈ B (4.18)SINRu (v) ≥ ρu, ∀u ∈ U (4.19)∑u∈U‖vbu‖22 ≤ Pb, ∀b ∈ B (4.20)∑b∈B∑u∈U∣∣gHbvvbu∣∣2 ≤ Iv, ∀v ∈ V (4.21)‖vbu‖22 ≤ xbu · Pb, ∀u ∈ U , ∀b ∈ B (4.22)where constraints (4.17)-(4.18) ensure that a base station is active if and only if it servessome users. In (4.19), ρu denotes the target SINR of user u. The above problem is hard tosolve due to the non-convexity of the SINR constraints and to its combinatorial nature.Let us create the matrix V =[vu1 , . . . ,vu|U|]∈ Cm|B|×|U| which concatenates all beam-forming vectors and define the following set Fv of beamforming vectors:864.4. Joint Clustering and Beamforming OptimizationFv =v ∈Cm|B|·|U|×1 :∥∥∥∥∥∥hHu V σu∥∥∥∥∥∥≤√1 + 1ρuhHu vu, ∀u ∈ URe(hHu vu)≥ 0, ∀u ∈ UIm(hHu vu)= 0, ∀u ∈ U∑u∈U ‖vbu‖22 ≤ Pmaxb , ∀b ∈ B∑b∈B∑u∈U∣∣gHbvvbu∣∣2 ≤ Iv, ∀v ∈ V(4.23)Thus, the following lemma tells us how to reduce the complexity of the problem.Lemma 4.1. By fixing the values of the clustering variables to(xk,yk), problem (4.14) canbe reduced to a convex beamforming optimization problem(Sk)defined by:variable vminimize f1 (v) + f2(xk,yk)(4.24)subject to v ∈ Fv (4.25)‖vbu‖22 ≤ xkbu · Pmaxb , ∀ (u, b) ∈ U × B (4.26)and the constraint set Fv defined in (4.23) is convex and compact.Proof. See Appendix C.14.4.4 Generalized Benders Decomposition MethodGiven the previous results, we will next derive an provably-convergent iterative algorithmfor solving problem (4.14). Using the Benders decomposition method [90], we will sepa-rate the clustering optimization and the cooperative beamforming design. Define H as theset of clustering variables for which the corresponding beamforming problem (S) admits afeasible solution:874.4. Joint Clustering and Beamforming OptimizationH ={(x,y) : ∃v ∈ Fv ‖vbu‖22 ≤ xbu · Pmaxb , ∀ (u, b) ∈ U × B}(4.27)Then, we can rewrite the original problem as a clustering master problem (M):(M)minimizex yg (x,y)subject to (x,y) ∈ H ∩ F(x,y)(4.28)For any arbitrary (x,y), the objective function g of problem (M) returns the optimal valueof the beamforming optimization (S) in (4.24). In (4.28), the set F(x,y) gathers the con-straints that are only related to the clustering variables as:F(x,y) =(x,y) :xbu ∈ {0, 1} , ∀ (u, b) ∈ U × Byb ∈ {0, 1} , ∀b ∈ Byb ≤∑u∈U xbu, ∀ (u, b) ∈ U × Byb ≥ xbu, ∀ (u, b) ∈ U × B(4.29)The equivalence between the original problem (4.14) and (4.28) is stated below.Proposition 4.1. If (x∗,y∗,v∗) is an optimal solution of the original mixed-integer problem in(4.14), then (x∗,y∗) is optimal for the clustering master problem (M) in (4.28). Conversely,if (x∗,y∗) solves problem (M) and if v∗ solves the beamforming problem (S) in (4.24) with(x,y) = (x∗,y∗), then (x∗,y∗,v∗) solves the original problem (4.14). Moreover, the originalproblem is infeasible if and only if (M) is infeasible.Proof. The proof of the above proposition follows directly from the partitioning proceduredescribed above and from the definitions of the function g and of the master problem(M).884.4. Joint Clustering and Beamforming OptimizationTo exploit the convexity of (S), we derive an iterative algorithm in which we solvethe subproblem (S) and update the clustering variables at each iteration. But the masterproblem (M) cannot be solved in its current form (4.28) since the function g is only knownimplicitly. To circumvent this, we will rewrite (M) in its dual form and apply the GBDmethod to solve it iteratively.Dual representation of the master problemLet us consider the subproblem (S) and define its partial Lagrangian with respect to thecomplicating constraints:L (v, λ;x,z) , f (v;x, z) +∑b∈B∑u∈Uλbu(‖vbu‖22 − xbu · Pb)(4.30)where the Lagrange multipliers satisfy λbu ≥ 0, ∀ (u, b) ∈ U × B.Moreover, we define a second type of Lagrangian function L with a parameter xk as:L(v, µ,xk),∑b∈B∑u∈Uµbu(‖vbu‖22 − xkbu · Pmaxb)(4.31)where the multiplier µ also satisfies µbu ≥ 0, ∀ (u, b) ∈ U × B. Note that any convexprogramming algorithm such as interior point methods can produce the optimal multipliervector λk along with vk without any extra computations [61]. Using these Lagrangian func-tions, we can obtain a more explicit reformulation of problem (M) by using the followingresult.Proposition 4.2. The master problem (M) in (4.28) is equivalent to the following problem(P ′):894.4. Joint Clustering and Beamforming Optimization(P ′)variables x y γminimize γ(x,y) ∈ F(x,y)γ ≥ ξ (x,y;λ) , ∀λ ≥ 00 ≥ ξ (x;µ) , ∀µ ≥ 0 :∑(b,u)∈B×U µbu = 1(4.32)where the support functions ξ and ξ are defined as the minimum of the Lagrange functionswith respect to v:ξ (x,y;λ) = minv∈FvL (v, λ,x,y) (4.33)ξ (x;µ) = minv∈FvL (v, µ,x) (4.34)Proof. See Appendix C.2.Although the equivalent formulation (P ′) is more explicit than (M), there are still twoissues with (P ′). First, the functions ξ and ξ defining constraints (4.32) are representedas inner minimization problems. Second, (P ′) involves an infinite number of constraintswhich makes it computationally impossible to solve. To overcome these, we first obtain anexplicit expression for each support function ξ and ξ.Lemma 4.2. If vk and λk are optimal primal and dual solutions for the convex subproblem(Sk), then the function ξ of (x,y), when parameterized by λk, is explicitly given by:ξ(x,y;λk)= L(vk, λk,x,y), ∀ (x,y) (4.35)Proof. See Appendix C.3.904.4. Joint Clustering and Beamforming OptimizationIn a similar way, we can also obtain an explicit expression of the function ξ as explainednext.Lemma 4.3. Suppose that(vk, αk)and µk are optimal primal and dual solutions for thefollowing convex feasibility problem(F k):(F k)variables v αminimize αsubject to v ∈ Fvα ≥ 0‖vbu‖22 − xkbu · Pmaxb ≤ α, ∀ (u, b) ∈ U × B(4.36)Then, the function ξ of x, when parameterized by µk, is explicitly given by:ξ(x;µk)= L(vk, µk,x), ∀x (4.37)and the optimal multiplier µk must satisfy∑(b,u)∈B×U µbu = 1.In the feasibility problem(F k)in (4.36), we relax the complicating constraints (4.22)and minimize the maximum constraint violation α. Note that(Sk)is infeasible if and onlyif the optimal value of(F k)is strictly positive. Next, we will use relaxation to solve themaster problem (P ′) iteratively with a finite number of constraints.Algorithmic procedureThe basic idea of the GBD method is to generate, at each iteration, an upper bound and alower bound on the optimum of the original problem until those bounds meet. The lowerbound is obtained by solving a relaxed version of problem (P ′) and the upper bound is914.4. Joint Clustering and Beamforming OptimizationInitialize assignmentvariablesSolveBeamformingSubproblem       Feasible?Generate a newOptimality CutGenerate a newFeasibility CutConvergenceTestYESYESNONOSolveMaster ProblemFeasible? NOReturnOptimalSolutionReturnInfeasibilitycerticateYESUpdateUpper BoundUpdateLower Boundandx ySk( )M k( )vuk( ) u U buk( ) b B,u USolveFeasibility Subproblem       Fk( )L v k , k ,x,y( )0 L v k ,µ k ,x( )vuk( ) u U µbuk( ) b B,u Uxk+1 yk+1Figure 4.2 – Generalized Benders Decomposition procedure for joint BS assignment and multi-cell beam-forming problem.computed from the optimal values of beamforming subproblems(Sk). The overall algo-rithm is presented in the flowchart in Figure 4.2.First, we start with an initial assignment x = x0 and y = y0. Next, we solve the sub-problem(Sk)which, when feasible, will give us an optimal pair(vk, λk)of beamformersand Lagrange multipliers. Then, we use these to generate the following optimality cutγ ≥ L(vk, λk,x,y). Otherwise, when subproblem(Sk)is infeasible, we solve the feasibil-ity problem(F k)and use the solution and multiplier(vk, µk)to generate a feasibility cut0 ≥ L(vk, µk,x). Depending on whether(Sk)is feasible or not, an optimality or feasibilitycut is added to the following relaxed version of the master problem (M) at each iterationk:924.4. Joint Clustering and Beamforming Optimization(Mk)variables γ,x,yminimize γsubject to (x,y) ∈ F(x,y)γ ≥ L(vl, λl,x,y), ∀l ∈ Ikfeas0 ≥ L(vj, µj,x), ∀j ∈ Ikinfeaswhere Ikfeas ={l ∈ {1, . . . , k} |(Sl)feasible}and Ikinfeas ={l ∈ {1, . . . , k} |(S l)infeasible}are the sets of iteration indices up to k for which the subproblems{(Sl)}l=1,...,kwere fea-sible or infeasible respectively.Solving the relaxed problem(Mk)will give new binary clustering variables(xk+1,yk+1)for the next iteration k+ 1. In addition, the optimal value γk of(Mk)gives a lower boundLBk = γk for the optimal value of the original problem. This lower bound is non-decreasingwith k because adding a new cut at each iteration can only shrink the feasibility set. Onthe other hand, the upper bound is updated at each iteration k as UBk , minl∈Ikfeasf(xl,yl,vl),which is clearly non-increasing with k. This iterative process is carried on until the dif-ference between these bounds is less than a prespecified tolerance  ≥ 0. From [90], thefinite convergence property of this algorithm is restated below.Proposition 4.3. For any given  ≥ 0, the GBD method algorithm terminates in a finitenumber of iterations.Proof. The proof directly follows Lemma 4.1 and from the results of Geoffrion in [90].Convergence acceleration techniques Although the optimal convergence of the Ben-ders’ algorithm is guaranteed, it can exhibit slow convergence in practice. In fact, the934.4. Joint Clustering and Beamforming Optimizationbinary assignment variables(xk+1,yk+1)obtained from(Mk)does not necessarily guaran-tee that(Sk+1)will be feasible at each iteration k. Another undesirable feature of cuttingplane algorithms such as Benders decomposition is that in early iterations, the solutionstend to oscillate from one region of the feasible set to another, thus slowing convergence[99]. Thus, we propose two techniques to speed up the convergence of the classical Ben-ders’ algorithm.• First, when the subproblem(Sk)becomes infeasible, we solve the feasibility problem(F k)and construct a feasibility cut as prescribed by the classical algorithm. Then,we check all pairs A of base station and user that create the maximum violation αˆ:A ={(b, u) ∈ B × U | xkbu = 0, ‖vbu‖22 − xkbu · Pb = αˆ}(4.38)After that, we select a new feasible assignment variables x by setting xbu = 1, ∀ (b, u) ∈A, solve the corresponding subproblem to get an extra optimality cut, and add thislater to the master problem with the feasibility cut. The idea is to restore the fea-sibility whenever we encounter an infeasible assignment and to generate as manyoptimality cuts as possible in the first iterations.• Second, we add valid cuts to the relaxed master problem to speed up the algorithm.Adding extra constraints shrinks the feasibility set and helps find feasible and moreefficient solution in early iterations. Using our knowledge of the clustering and co-operative beamforming problem, we add the following constraints:∑b∈B xbu ≥ , ∀u ∈ Uxb¯uu ≥ xbu, ∀b ∈ B\{b¯u}∀u ∈ UThe first constraint enforces that each user must be served by some base stationswhereas the second one ensures that the base station b¯u that has the strongest chan-944.5. Distributed Multi-cell Cooperative Beamformingnel to a user u first serve that user. Only when cooperation is required, the other basestations can jointly serve user u with b¯u.4.5 Distributed Multi-cell Cooperative BeamformingIn this section, we propose a decentralized algorithm for cooperative beamforming whenthe clustering is fixed. In fact, the centralized optimization is impractical since it requiresthe exchange of global channel information over the backhaul and may result in largeoverhead [86]. Instead, our proposed distributed algorithm requires only local channelinformation and some limited signaling.4.5.1 Problem ReformulationDenote by C the set of all clusters in the small-cell system. Each cluster c has Bc ⊆ Bas its set of cooperating BSs that jointly serve the users Uc ⊂ U . To simplify the designof a decentralized algorithm, we assume that the clusters do not overlap, i.e. we haveBc ∩Bc′ = ∅ and Uc ∩Uc′ = ∅ for each pair of clusters (c, c′) ∈ C2. With this assumption, thereceived signal for user u served by a cluster c becomes:yu = (hcu)H vcusu︸ ︷︷ ︸Desired signal+∑j∈Uc\{u}(hcu)H vcjsj︸ ︷︷ ︸Intra-cluster interference+∑c′∈C\{c}∑k∈Uc′(hc′u)Hvc′k sk︸ ︷︷ ︸Inter-cluster interference+ nu (4.39)where hcu , (hbu)b∈Bc ∈ Cm|Bc|×1 and vcu , (vbu)b∈Bc ∈ Cm|Bc|×1.Our goal is to make the beamforming optimization separable between the clusters. Tothis end, we first transform the SINR of each user u as:SINRu =∣∣∣(hcu)H vcu∣∣∣2∑j∈Uc\{u}∣∣∣(hcu)H vcj∣∣∣2+∑c′∈C\{c} z2c′u + σ2u(4.40)by replacing the inter-cluster interference term received by u from each c′ ∈ C\ {c} with aslack variable z2c′u ≥ 0. To keep the formulation coherent, we also need the constraints:954.5. Distributed Multi-cell Cooperative Beamforming∑k∈Uc′∣∣∣∣(hc′u)Hvc′k∣∣∣∣2≤ z2c′u, ∀(c′, u)∈ C\ {c} × Uc, ∀c ∈ C (4.41)Note that each variable zc′u exactly couples a pair of clusters: user u’s serving cluster cand its interfering cluster c′. Thus, the interference constraints (4.41) are equivalent to:∑l∈Uc∣∣∣(hcj)H vcl∣∣∣2≤ z2cj , ∀j ∈ U\Uc, ∀c ∈ C (4.42)Despite the equivalence, having constraints (4.41) at each cluster c is not useful becausecluster c cannot control the interference power z2c′u received by its user u ∈ U . Instead,cluster c can control the interference power z2cj that it creates to a non-intended userj ∈ U\Uc through (4.42). To simplify the notation, we collect all coupling inter-clusterinterference variables into z , (zcj)j∈U\Uc, c∈C ∈ R(|C|−1)|U|+ .Similarly, we create a new slack variable ic , (icv)v∈V ∈ R|V|+ that collects the interfer-ence temperature created by cluster c to each primary user. Then, we decompose the totalinterference temperature constraints (4.21) as follows:∑l∈Uc∣∣∣(gcv)H vcl∣∣∣2≤ icv, ∀v ∈ V , ∀c ∈ C∑c∈C icv ≤ Iv, ∀v ∈ V(4.43)As a result, the beamforming optimization problem becomes:964.5. Distributed Multi-cell Cooperative Beamformingvariables (vc)c∈C , z, (ic)c∈Cminimize∑c∈C∑k∈Uc‖vck‖2subject to SINRu(vc, (zc′u)c′∈C\{c})≥ ρu, ∀u ∈ Uc, ∀c ∈ C∑l∈Uc∣∣∣(hcj)Hvcl∣∣∣2≤ z2cj, ∀j ∈ U\Uc, ∀c ∈ C∑l∈Uc∣∣∣(gcv)H vcl∣∣∣2≤ icv, ∀v ∈ V , ∀c ∈ C∑l∈Uc‖vbl‖22 ≤ Pmaxb , ∀b ∈ Bc, ∀c ∈ C∑c∈C icv ≤ Iv, ∀v ∈ V(4.44)Adding local interference variables and consistency constraintsIn (4.44), the clusters are still coupled through the variables z, {ic}c∈C and the sharedconstraints∑c∈C icv ≤ Iv, v ∈ V. To remove the coupling through z, we create a localvariable z˜c for each cluster c, which is defined by z˜c , (z˜rxc , z˜txc ) ∈ R(|C|−2)|Uc|+|U|+ . Thus, itcontains only the following inter-cluster interference variables relevant to cluster c:z˜rxc , (z˜rxc′u)(c′,u)∈C\{c}×Uc ∈ R(|C|−1)|Uc|+ (4.45)z˜txc ,(z˜txcj)j∈U\Uc∈ R(|U|−|Uc|)+ (4.46)The local variable z˜rxc′u of cluster c denotes the desired interference that cluster c wants itsuser u ∈ Uc to receive from an interfering cluster c′ whereas the local variable z˜txcj of clusterc denotes the interference that cluster c desires to transmit to a non-intended user j. Toenforce a consistency between the local and coupling variables, we add the constraints:z˜c = Bcz, ∀c ∈ C (4.47)974.5. Distributed Multi-cell Cooperative Beamformingwhere Bc is a binary matrix that maps the local z˜c to the part of z relevant to cluster c.Similarly, we introduce local interference temperature variable i˜c for each c with theconsistency constraints:i˜c = ic, ∀c ∈ C (4.48)After this step, the beamforming problem (4.24) is equivalent to:variables(vc, z˜c, i˜c)c∈C, z, (ic)c∈Cminimize∑c∈C∑k∈Uc‖vck‖2subject to(vc, z˜c, i˜c)∈ Fc, ∀c ∈ C∑c∈C icv ≤ Iv, ∀v ∈ Vz˜c = Bcz, ∀c ∈ Ci˜c = ic, ∀c ∈ C(4.49)In (4.49), the local variables(vc, z˜c, i˜c)belongs to a convex set Fc defined by:984.5. Distributed Multi-cell Cooperative BeamformingFc =(vc, z˜c, i˜c):∥∥∥∥∥∥(hcu)H Vc ζ˜u σu∥∥∥∥∥∥≤√1 + 1ρu (hcu)H vcu, ∀u ∈ UcRe((hcu)H vcu)≥ 0, ∀u ∈ UcIm((hcu)H vcu)= 0, ∀u ∈ Uc∑l∈Uc∣∣∣∣(hcj)Hvcl∣∣∣∣2≤ z˜2cj , ∀j ∈ U\Uc∑l∈Uc∣∣∣(gcv)H vcl∣∣∣2≤ i˜cv, ∀v ∈ V∑l∈Uc ‖vbl‖22 ≤ Pmaxb , ∀b ∈ Bc(4.50)in which the matrix Vc ,(vcj)j∈Uc∈ Cm|Bc|×|Uc| concatenates the beamforming vectorsof cluster c and the vector ζ˜u , (z˜c′u)c′∈C\{c} ∈ R1×(|C|−1)+ collects the local variables ofthe interference received by its user u. The new formulation (4.49) is obtained afterusing the manipulations in (4.40)-(4.43), introducing the local variables and consistencyconstraints (4.47)-(4.48), and rewriting the SINR constraints in a conic form. As a result,the equivalent beamforming problem (4.49) is now convex and quasi-separable.4.5.2 Alternating Direction Method of Multiplier (ADMM)Next, we show how to solve (4.49) using the ADMM method. First, let us form the aug-mented Lagrangian Laug :Laug((vc, z˜c, i˜c)c∈C, z, (ic)c∈C)=∑c∈C∑k∈Uc‖vck‖2 + pICc (Bcz, z˜c) + pITc(ic ,˜ic) (4.51)where the penalty functions pICb and pITb in (4.51) are defined by:pICc (Bcz, z˜c) , λ>c (Bcz−z˜c) +ωIC2‖Bcz−z˜c‖22 (4.52)pITc(ic ,˜ic), µ>c(ic−˜ic)+ωIT2∥∥∥ic−˜ic∥∥∥22(4.53)994.5. Distributed Multi-cell Cooperative BeamformingThe Langrangian dual variables λc = [λrxc , λtxc ] ∈ R(|C|−2)|Uc|+|U| and µc ∈ R|V| in (4.52) and(4.53) play the role of consistency prices, and ωIC, ωIT ≥ 0 are fixed penalty weights for thequadratic regularization terms.Using the ADMM method, we can solve problem (4.49) by sequentially updating thelocal variables(vc, z˜c, i˜c)c∈C, the coupling interference variables(z, (ic)c∈C)and the dualvariables (λc, µc) at each iteration n with the following three steps:(S.1) Update the local variables:(v(n+)c , z˜(n+)c , i˜(n+)c)c∈C= argmin(vc,z˜c ,˜ic)∈Fc, ∀c∈CLaug((vc, z˜c, i˜c)c∈C, z(n),(i(n)c)c∈C)(4.54)(S.2) Update the coupling variables:(z(n+),(i(n+)c)c∈C)= argminz,(ic)c∈CLaug((v(n+1)c , z˜(n+1)c , i˜(n+1)c)c∈C, z, (ic)c∈C)(4.55)subject to∑c∈Cicv ≤ Iv, ∀v ∈ V (4.56)(S.3) Update the consistency prices:λ(n+ 1)c = λ(n)c + ωIC(Bcz(n+1)−z˜(n+1)c)(4.57)µ(n+ 1)c = µ(n)c + ωIT(i(n+1)c −˜i(n+1)c)(4.58)Given the convexity of problem (4.49), the convergence properties of the ADMM algo-rithm are given as follows.Proposition 4.4. If problem (4.49) has a non-empty feasible set, then the ADMM-basedalgorithm converges as follows: (i) the total transmit power converges to the optimal value P ∗Tas n→∞; (ii) a consensus will be reached between the clusters, i.e. Bcz(n)−z˜(n)c → 0, ∀c andi(n)c −˜i(n)c → 0, ∀c as n → ∞, and (iii) the iterates(λ(n)c , µ(n)c)will converge to the optimalmultipliers of problem (4.49).1004.5. Distributed Multi-cell Cooperative BeamformingThis result follows from the convexity of problem (4.49). Assuming that the probleminstance is feasible and Slater’s constraint qualification holds, then problem (4.49) satisfiesthe required assumptions under which ADMM is guaranteed to converge [100].4.5.3 Decentralized Beamforming and Signaling SchemesThe previous ADMM-based algorithm can be carried out in a decentralized way by allowingthe clusters to exchange some signaling information. First, the optimization problem instep (S.1) is completely separable. Thus, each cluster c can independently update itsbeamformers and local interference variables.For the step (S.2) in (4.55), z and (ic)c∈C can be updated separately with:z(n+1) =12∑c∈CB>c(z˜(n+1)c −1ωICλ(n)c)(4.59)and the following quadratic program as:(i(n+1)c)c∈C= argmin∑c∈C icv≤Iv , v∈V∑c∈Cµ>c ic +∑c∈CωIT2∥∥∥ic−˜i(n+1)c∥∥∥2(4.60)Next, the previous z-update and i-update can be carried out locally at each cluster. Infact, it can be shown that the z-update for cluster c is done component-wise as follows:z(n+1)c′u =12(z˜rx(n+1)c′u −1ωICλrx(n)c′u)+12(z˜tx(n+1)c′u −1ωICλtx(n)c′u), ∀u ∈ Uc, ∀c′ ∈ C\ {c} (4.61)z(n+1)cj =12(z˜tx(n+1)cj −1ωICλtx(n)cj)+12(z˜rx(n+1)cj −1ωICλrx(n)cj), ∀j ∈ Uc′ , ∀c′ ∈ C\ {c} (4.62)For this to be possible, each pair of clusters (c, c′) needs to exchange some signaling infor-mation. Cluster c′ needs to send to cluster c the real vectors(z˜tx(n+1)c′u −1ωICλtx(n)c′u)u∈Ucand(z˜rx(n+1)cj −1ωICλrx(n)cj)j∈Uc′and vice versa. Similarly, the coupling variable (ic)c∈C can be lo-cally updated at each cluster with (4.60) after having each pair of clusters (c, c′) exchangetheir local variables(i˜(n+1)c , i˜(n+1)c′)∈ R2|V|.For step (S.3), each cluster c can update the consistency prices (λc, µc) independentlyfrom (4.57)-(4.58) with no extra signaling cost. As a result, the total signaling overhead1014.5. Distributed Multi-cell Cooperative Beamformingper iteration required for achieving optimality is:∑c∈C∑c′∈C\{c}(|Uc|+ |Uc′ |+ |V|) = 2 (|C| − 1) (|U|+ |V|) (4.63)In practice, it is however desirable to limit the signaling and aim for a near-optimal per-formance. To do that, we first limit the number of iterations as done in [101]. Specifically,the clusters can agree to fix their local interference to the consented values(Bcz(n+1), i(n+1)c)c∈Cafter step (S.2) of an intermediate iteration n. Then, they recompute their beamformersas: minimizevc∑k∈Uc ‖vck‖2subject to(vc,Bcz(n+1), i(n+1)c)∈ Fc(4.64)This extra computation (4.64) enables to obtain a feasible solution at an intermediate it-eration provided that the consented values(Bcz(n+1), i(n+1)c)c∈Crender the subproblems(4.64) feasible for all clusters. Otherwise, they need to carry more iterations. Our sim-ulation results however shows that tens of iterations are enough to achieve near-optimalperformance.Second, we reduce the signaling overhead per iteration through user grouping. Basedon the local channel information {hcu}u∈Uc (and/or {gcv}u∈V), each cluster c can partitionthe small-cell users (and/or the primary users) into two groups as Uc = U1c ∪ U2c (and/orV = V1c ∪V2c ). The users in the first group U1c are those whose desired channel gains from care weaker compared to the users in the second group U2c . On the other hand, the primaryusers in V1c are those whose interference channels are stronger compared to V2c . Usingdifferent restriction rules, the following limited signaling schemes are proposed:• Fractional signaling: Since U1c and V1c are more vulnerable to interference, the frac-tional signaling scheme does not impose any restriction on their interference vari-ables. Instead, each cluster c restricts the inter-cluster interference (and/or interfer-ence temperature) to be equal for all small-cell (and/or primary) users in the second1024.5. Distributed Multi-cell Cooperative BeamformingTable 4.2 – Overhead per iteration of limited signaling schemes.SignalingschemesSignaling restriction on inter-clusterinterference onlySignaling restriction on inter-cluster interferenceand interference temperatureFractional∑c∈C∑c′∈C\{c}(∣∣U1c∣∣+∣∣U1c′∣∣+ |V|+ 2) ∑c∈C∑c′∈C\{c}(∣∣U1c∣∣+∣∣U1c′∣∣+∣∣V1c∣∣+∣∣V1c′∣∣+ 4)Group-wise∑c∈C∑c′∈C\{c} (4 + |V|) 16 (|C| − 1)Cluster-wise∑c∈C∑c′∈C\{c} (2 + |V|) 8 (|C| − 1)group U2c (or V2c ). This can be achieved by adding extra constraints in the optimiza-tion problem of step (S.1) and by setting the initial consistency prices to be equalfor the users in the second group. With that, it can be verified by induction that forevery iteration n > 0, the signaling information exchanged between (c, c′) becomeequal for all small-cell users in U2c (and/or all primary users in V2c ), thereby reducingthe overhead.• Group-wise signaling: As before, each cluster c partitions its users into two groups.However, the inter-cluster interference (or interference temperature) of the small-cell(or primary) users are restricted to be equal in each of the groups.• Cluster-wise signaling: Here, each cluster c does not divide the primary or small-cellusers into two groups but simply enforces the interference received by all small-cellusers and/or all primary users to be equal.The overhead per iteration of these signaling schemes is summarized in Table 4.2. Forcomparison, the total overhead of the central optimization due to the exchange of globalchannel information is given by∑c∈C∑c′∈C\{c} 2m · |Bc′| (|U|+ |V|) assuming that eachcomplex channel coefficient is sent over the backhaul as two real scalars.1034.6. Simulation ResultsTable 4.3 – Power cost and system parameters in Chapter 4.Simulation parameters ValuesNumber of antenna at base stations m 3Number of base stations 7Inter-base station site distance 500mMaximum transmit power, Pmax 46dBmSystem bandwidth 5MHzChannel bandwidth 200KHzThermal noise power density −174dBm/HzPower amplifier efficiency, µPA 35%Static power cost, Pstatic 6.8WPower supply battery backup, CPS 11%Dynamic signal processing power, psp 5.8WPower efficiency of backhaul 0.1W/Mbps4.6 Simulation ResultsNext, we simulate a two-tier network with a total of seven base stations. We assumethat the small cells are the secondary users and they share the spectrum with the primarymacro-cell users. Then, we apply the cognitive resource allocation framework presentedabove to this network. As a result, we use the terms macro-cell user and primary userinterchangeably. Similarly, small-cell user and secondary user are used interchangeably inthis section. All these users are uniformly distributed within a circular area with a radius of1km from the center base station. Our simulation parameters are listed in Table 4.3. Themaximum base station transmit power, the cooperative processing power and fixed con-sumed power parameters are normalized to the subcarrier bandwidth. In our simulations,we solved the convex beamforming subproblems using the convex optimization softwareCVX [1].1044.6. Simulation Results−1 −0.5 0 0.5 1−1−0.500.51(a) 7 small-cell userswith SINR 10dB−0.5 0 0.5 1−1−0.500.51(b) 7 small-cell users with SINR20dB−1 −0.5 0 0.5 1−1−0.500.51(c) 10 small-cell users with SINR10dBFigure 4.3 – Optimal clustering for different network topologies and SINR targets4.6.1 Performance of Joint Clustering and BeamformingIn Figure 4.3, we illustrate the optimal clustering for two different network topologies.The primary users are denoted by squares whereas small-cell users by circles. The totalinterference power limit at each primary user is Iv = 3dB above the noise level. In Figure4.3(a), the simple inter-cell coordination scheme is optimal when |U| = |V| = 7 andρ = 10dB. However, some BS cooperation is required to support a higher SINR targetρ = 20dB (Figure 4.3(b)) or to serve |U| = 10 small-cell users (Figure 4.3(c)). Thus, theflexible cooperation scheme allows the BSs to share their antennas only when they needto jointly serve some cell-edge users with a high quality of service or when their limitedspatial degrees of freedom do not allow them to individually serve more small-cell userswhile protecting the primary users.For the network topology in Figure 4.3(a), we compare the performance and powercost of the flexible cooperation with those of inter-cell coordination and full cooperationschemes in Figure 4.4. While the simple inter-cell coordination is optimal at low SINR,it is no longer feasible when ρ > 10dB. We also observe from Figures 4.4(a)-(b) that fullcooperation achieves the lowest transmit power thanks to the macro-diversity gain. Butsince the cooperative power cost is high, significant energy can be saved by resorting to the1054.6. Simulation Results0 5 10 15 20 250510152025SINR of Small-cell Users (dB)Total Power Cost (Watts)  Inter−cell CoordinationFlexible CooperationFull Cooperation(a) Total power consump-tion0 5 10 15 20 25−15−10−50510152025SINR of Small-cell Users (dB)BS Transmit Power (dB)  Inter−cell CoordinationFlexible CooperationFull Cooperation(b) Total transmit power0 5 10 15 20 25−6−4−2024681012SINR of Small-cell Users (dB)Total Interference to primary users (dB)  Inter−cell CoordinationFlexible CooperationFull Cooperation(c) Total interference temperaturepowerFigure 4.4 – Performance comparison and effect of SINR requirements of small-cell users.0 5 10 15 20 250102030405060708090100Number of Small-cell UsersInfeasibility rate (%)    Inter-cell Coord. ρ = 5dB  Inter-cell Coord. ρ = 10dB  Inter-cell Coord. ρ = 20dB  Flexible Coop. ρ = 5dB  Flexible Coop.   Flexible Coop. ρ = 20dB  Full Cooperation ρ = 5dB  Full Cooperation ρ = 10dB  Full Cooperation ρ = 20dBρ = 10dBFigure 4.5 – Effect of the number of small-cell users with 10dB SINR targetflexible cooperation. Figure 4.4(c) shows that the interference to primary users increaseswith the SINR target until the limit is reached.In Figure 4.5, we present the effect of the number of small-cell users on the feasibilityof supporting the primary and small-cell users’ requirements. Here, we have |V| = 7primary users. For a given number of secondary users, we generate 1000 random networkinstances with different user locations and channels. With |U| = 10 and ρ = 10dB, only15% infeasible cases were encountered when cooperation was used. Otherwise, 65% of theproblem instances were infeasible with the simple inter-cell coordination scheme. Thus,1064.6. Simulation Results0 10 20 30 40 50 60 70 80 90 100 110 120−50510152025IterationsTotal power cost (Watts)  Improved Benders algorithmClassical Benders algorithmUpper BoundsLower Bounds(a) Classical and improved Benders algo-rithms20 40 60 80 100 120 140 160 180 20010−410−310−210−1100IterationNormalized Transmit Power Accuracy              ρ = 0dBρ = 6dBρ = 10dB(b) ADMM-based decentralized algo-rithmFigure 4.6 – Convergence of proposed algorithmsthe flexible cooperation scheme enhances the capabilities of the base stations to combatinterference. Although it has the same infeasibility rate as the full cooperation scheme, itcan also save power by reducing the degree of cooperation whenever possible.4.6.2 Convergence of Proposed AlgorithmsNext, we analyze the convergence of our proposed algorithms for scenario in Figure 4.3(c).For the joint clustering and beamforming problem, we show the convergence of the Ben-ders algorithm when optimizing the clustering and beamforming for the scenario in Figure4.3(c). We see that the convergence rate of the algorithm is largely improved by applyingfeasibility restoration and by inserting valid cuts. In fact, those two simple techniques en-abled to strengthen the lower bounds and generate as many optimality cuts as possible inthe early iterations. The typical convergence of the ADMM-based algorithm is shown inFigure 4.6(b). Note that the normalized power accuracy is defined as |∑c∈C∑b∈BcPb−P ∗T |P ∗T.1074.6. Simulation Results−1 −0.5 0 0.5 1−1−0.500.51Figure 4.7 – Cooperative network with 3 clusters and |U| = 14, |V| = 3.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2051015Channel realization indexTotal Transmit Power (dB)   Optimal schemeFractional signaling − 50 itersGroup−wise signaling − 50 itersCluster−wise signaling − 50 itersFractional signaling − 10 itersGroup−wise signaling − 10 itersCluster−wise signaling − 10 iters(a) |V| = 3 and ρ = 6dB1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2051015  Channel realization indexTotal Transmit Power (dB) Optimal schemeFractional signaling − 50 itersGroup−wise signaling − 50 itersCluster−wise signaling − 50 itersFractional signaling − 10 itersGroup−wise signaling − 10 itersCluster−wise signaling − 10 iters(b) |V| = 6 and ρ = 3dBFigure 4.8 – Performance comparison of limited signaling schemes with I = 3dB.1084.6. Simulation Results4.6.3 Performance of Decentralized Scheme with Limited SignalingWe consider the network in Figure 4.7 with three clusters, |B| = 7, |U| = 14, ρ = 6dBand Iv = 3dB. We compare the performance of the decentralized schemes with limitedsignaling for different channel realizations. We use the extra computation in (4.64) toobtain a feasible solution after 10 and 50 iterations. In Figure 4.8(a), we have |V| = 3and the signaling restriction is imposed only on the inter-cluster interference. In Figure4.8(b), we increase the number of primary users to |V| = 6 and add restriction on theinterference temperature. Each cluster partitions the users into two groups of equal sizefor the fractional and group-wise signaling schemes. Figures 4.8(a) and (b) show thatthe fractional signaling, followed by group-wise signaling, perform near the optimum. Onthe other hand, the cluster-wise signaling creates a large optimality gap especially whenthe channels are badly conditioned. Finally, using (4.63) and Table 4.2, we found thatthe fractional and group-wise signaling can reduce the overhead per iteration by about30% and 60% respectively compared to the optimal ADMM scheme while achieving near-optimal performance. When compared to the centralized optimization, the total overheadis reduced by about 67% and 80% respectively with 10 iterations.109Chapter 5Summary, Conclusions and Future WorkIn order to support the exploding mobile data traffic, wireless operators must drasticallyincrease the network capacity through an ultra-densification of the network. A huge chal-lenge for dense heterogeneous deployments is to keep the energy consumption at thesame level as today’s. In this thesis, we studied different resource allocation and coop-eration schemes for enabling energy-efficient wireless heterogeneous two-tier networks.In particular, we focused on the coordinated multi-cell power allocation, energy coopera-tion and cooperative multi-point beamforming in such networks. Throughout this thesis,we leveraged different convex optimization techniques to solve these non-convex resourceallocation problems. Our proposed algorithms are efficient, provably-convergent and ap-plicable to various heterogeneous network deployments. In the following, we present asummary of the contributions made in this thesis along with some concluding remarks.5.1 Summary and ConclusionsIn Chapter 2, we proposed a multi-cell coordination framework for maximizing the energyefficiency of wireless heterogeneous networks. To enable the coordination between thecells, the channel state information needs to be shared through some backhaul links. Al-ternatively, a centralized RAN architecture can also facilitate this coordination by havinga central unit controlling the power allocation of the remote radio units. When develop-ing energy-efficient power allocation algorithms, we considered two different scenarios:orthogonal multiple access and full spectrum sharing. Even with orthogonal transmis-1105.1. Summary and Conclusionssions, the resulting optimization problem is not convex and cannot be solved by stan-dard algorithms. Nonetheless, we derived a useful convex parameterization to handle thenon-convexity. Specifically, an efficient solution of the non-convex energy-efficiency max-imization problem is found by solving a sequence of convex parametric problems instead.Moreover, we showed that the optimal parameters of this surrogate problems turn out tobe the unique root of a non-linear system of equations. As a result, an algorithm based ona damped Newton method is derived and proved to converge to the global optimum. Inaddition, we also solved the general scenario of non-orthogonal transmissions, i.e. whenthe cells fully share the spectrum. This setup is more difficult due to the multi-user interfer-ence. Thus, another algorithm was derived using the minorization-maximization principleand it was shown to monotonically converges to at least a local optimum. These proposedoptimization algorithms can be used with various power constraints such as individualtransmit power, total power constraint and interference constraints. Moreover, we showedthat the proposed framework can be useful even with the presence of probabilistic interfer-ence constraints. Our simulation results validated the convergence and effectiveness of theproposed algorithms. By comparing the proposed scheme with a baseline non-cooperativescheme, we found that a simple coordination between the cells can drastically improve theenergy efficiency of heterogeneous networks. In fact, the achieved energy efficiency in-creases with the number of active small cells. These results confirm the need for multi-cellcoordination to achieve an efficient operation of dense heterogeneous networks.In Chapter 3, we studied the energy allocation problem in hybrid-powered wirelesstwo-tier networks. The dense deployment of base stations in future networks pose atremendous challenge to the energy cost. Therefore, it is desirable to use advanced yetsustainable energy solutions. Specifically, we assumed that each cell has a hybrid energysource and an energy storage system. The hybrid power source allows each cell to drawenergy from both renewable source and the conventional power grid. With this setup,we formulated an optimization problem, in which the cells maximize the sum of their1115.1. Summary and Conclusionsenergy efficiency by allocating their energy over a finite time horizon. Further, each cellimposes an average sum rate constraint. In contrast to Chapter 2, the new problem for-mulation takes into account the possible fluctuations of renewable energy arrivals at thecells. Our objective was to investigate the benefits of the hybrid power source in cellulartwo-tier networks. Thus, we focused on offline optimization and assumed a non-causalknowledge of the channel conditions and renewable energy arrivals. The algorithms pro-posed in Chapter 2 were extended to handle the non-convex average cell rate constraints.Whenever the new algorithm starts with a feasible power allocation, the subsequent iter-ates necessarily satisfy the rate constraints and monotonically converges to at least a localoptimum. Furthermore, we studied the joint energy allocation and energy cooperation inwireless two-tier networks. Precisely, we extended the energy-saving capability of the net-work by allowing the cells to exchange their harvested energy through a smart-grid powerinfrastructure. In this case, we propose to exploit both the spatial and time diversity ofthe renewable energy arrivals across the cells. Our numerical results showed that boththe hybrid power source and the energy cooperation scheme can significantly improve theenergy efficiency of cellular two-tier networks. A much higher gain can be obtained whenthey are used together. Hence, energy cooperation is a viable approach for reducing boththe operational and environmental costs of future heterogeneous networks. Nonetheless,there are still many practical issues that need further investigation to fully realize suchpossibility. Some of these open problems are discussed in the next section.In Chapter 4, we investigated the joint optimization of the clustering and cooperativebeamforming for minimizing the total energy cost in clustered two-tier systems. In con-trast to Chapter 2 and Chapter 3, we assumed here that the base stations can have multipleantennas. In addition, they cooperate by sharing their antennas and by jointly transmittingto their users. To achieve this form of cooperation, the cells must be connected by back-haul links through which they exchange the user data and/or signaling information. Theadvantages of using this coordinated multi-point transmission scheme are the robustness1125.2. Future Research Directionsagainst interference and the spatial multiplexing gain. However, this cooperation schememay incur some significant energy cost due to the multi-cell processing and the backhaulsignaling. Therefore, we propose a flexible cooperation scheme, in which the cooperativecluster for each user is optimized to adapt the degree of cooperation of the base stations.In fact, most existing works in the literature have overlooked this trade-off. Instead, weexplicitly modeled this optimization problem as a mixed-integer program and solved itusing the generalized Benders decomposition method. Precisely, we divided the probleminto a beamforming subproblem and a clustering master problem. The proposed Bendersalgorithm optimally converges in a finite number of iterations but it can exhibit a slowconvergence. Therefore, we derived simple techniques to accelerate its convergence. Wealso proposed a decentralized cooperative beamforming algorithm when the clusters arefixed. In fact, the use of distributed algorithms is of practical importance when the cellsare not connected through a central unit. With this distributed coordination, the clusterswere able to manage their interference through consensus. Given the limited capacity ofthe backhaul links, it is also important to limit the signaling overhead. By employing somelimited signaling schemes, it is possible to reduce the total overhead by about 30% and60% respectively at the cost of a marginal increase in total energy consumption. Moreimportantly, we showed that it is possible to achieve an efficient trade-off between energycost and performance when employing a coordinated multi-point transmission in hetero-geneous networks.5.2 Future Research DirectionsAs a result of the work done in this thesis, we present some possible directions of futureresearch as described in the following sections.1135.2. Future Research DirectionsDistributed algorithms for energy-efficient device-to-device networksThe optimization framework presented in Chapter 2 is suitable for heterogeneous networksthat have a centralized architecture. An example of this is a cloud-RAN based networksin which the cells are served by remote radio units that are connected to a central unitthrough optical fronthaul links. Alternatively, the centralized energy-efficiency optimiza-tion can also be performed for distributed cells under the condition that they can exchangetheir channel information through some backhaul links. Nonetheless, it is preferable if notnecessary in certain scenarios to employ distributed algorithms instead of a centralizedone. For instance, device-to-device communication systems do not have a centralized unitnor backhaul links that connect the devices. In such case, they need to coordinate theirresource allocation in a self-organized manner. Clearly, distributed algorithms will need toincur some signaling overhead to enable the devices or cells to achieve good performance.When designing distributed algorithms, an important criteria is to achieve a good trade-offbetween signaling overhead and performance. For this purpose, the centralized algorithmproposed in Chapter 2 can serve as a benchmark. Moreover, the computational and run-time complexities of the distributed algorithms should be at least comparable to that of acentralized one. This should not be too hard if a parallel update method of the Jacobi typeis employed [91]. Nonetheless, it is a challenging yet interesting problem to derive parallel,convergent and efficient algorithms for the non-convex energy-efficiency maximization. Toour knowledge, only the algorithm in [56], which converges to a Nash equilibrium, hasattempted such parallel updates for this problem. Nonetheless, we showed in Chapter 2 ofthis thesis that the achieved Nash equilibrium is not efficient for this problem. In addition,the distributed algorithm should be able to handle individual as well as shared power con-straints. Finally, another technical challenge is to analyze the convergence of the designeddistributed algorithm under asynchronous updates. This is an important practical issue fordevice-to-device networks given the lack of centralized coordination.1145.2. Future Research DirectionsStochastic energy-efficiency maximization with imperfect channelinformationIn Chapter 2, we addressed the possibility to have imperfect channel information betweenthe cells and the co-existing primary users in the system. For this, we proposed to useprobabilistic constraints and handle the tractability issue through robust approximationmethods. However, it is also possible that the channel state information between thecells themselves are uncertain. In this case, the uncertainty is present in the utility func-tion and our new goal is to maximize the expected energy efficiency. However, even ifthe exact probability distribution of the channels is known, it is intractable to obtain aclosed-form expression of such expected utility function. In this case, it is more practi-cal to use a stochastic approximation approach [102]. In this line, a method was recentlyproposed in [103] for stochastic non-convex rate maximization problems. Based on par-tial linearization [104] and a gradient averaging approach [105], the algorithm consistsof solving a sequence of simpler surrogate subproblems. It was shown to converge to astationary point [103]. However, the approximate objective function of the subproblemsneed to be strongly convex. Unfortunately, it is not easy to find such approximation forthe non-convex energy-efficiency function. Nonetheless, the application of stochastic ap-proximation and averaging methods is a promising direction for solving the stochasticenergy-efficiency maximization problem.Online adaptive optimization for joint energy allocation and energycooperationAnother interesting open problem is to design online algorithms for the joint energy allo-cation and energy cooperation problem. In practice, only information about past channelconditions as well as past/current energy arrivals are available at each cell. Therefore, anew framework is needed to solve this optimization problem in an adaptive manner. For-1155.2. Future Research Directionstunately, there is a growing interest for online algorithms not only for communications butalso for different applications such as smart-grid and machine learning on data streams. Toour knowledge, some of the frameworks that were previously used to design adaptive algo-rithms are dynamic programming [106], Lyapunov optimization [107] and online convexoptimization [108]. The traditional approach based on dynamic programming is difficultto apply here because the algorithm may suffer from the curse of dimensionality [106]. Onthe other hand, the algorithms derived from online convex optimization framework (e.g.mirror descent [108]) first need to be extended to handle the non-convexity of the energyefficiency function. Moreover, it should be able to satisfy both the time-varying energyconstraints and the average rate constraints. Therefore, further research efforts will beneeded to tackle this interesting problem.Energy-efficient load balancing and adaptive clustering forcooperative small-cell networksWith regard to Chapter 4, we assumed that each user has a fixed quality of service require-ment. This is reasonable if the amount of data traffic created by the users is slowly varyingand predictable. A more applications will be moved to the mobile cloud, future wirelessnetworks will have to manage a fast-changing and highly volatile data traffic across thenetwork. Cooperation between the small-cells would be needed in order to balance thedata load and achieve a satisfactory level of service for the users. Hence, the joint cluster-ing and resource allocation problem, studied in Chapter 4, can be extended to take intoaccount the load balancing of the cells. A new model and a new utility function will beneeded to capture the trade-off between energy cost and service delay.116Bibliography[1] M. Grant and S. 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Shamai, “Linear precoding via conic optimization forfixed MIMO receivers”, IEEE Transactions on Signal Processing, vol. 54, no. 1, pp. 161–176, 2006.130Appendices131Appendix AProof of Results in Chapter 2A.1 Proof of Proposition 2.1To prove Proposition 2.1, let us define the following function:L (θ,p;α,w, λ) = γ∑c∈Cθc +∑c∈Cαc (rc (pc)− θcCc (pc)) +∑l∈Vλlhl (p) , (A.1)which is parametrized by (γ, α, λ), with λ , (λl)l∈V .Proof. Suppose that(p̂, θ̂)is an optimal solution of the original problem (P ) formulatedin (2.10). Then, using the Fritz-John optimality conditions [109], there exists a triple(γ, α, λ)that satisfies:∂L∂p=∑c∈Cαc(∇pRc (p̂c)− θ̂c∇pCc (p̂c))+∑l∈Vλl∇phl (p̂) = 0, (A.2)∂L∂θi= γ − αcCc (p̂c) = 0, ∀c ∈ C, (A.3)αc(Rc (p̂i)− θ̂cCc (p̂c))= 0, ∀c ∈ C, (A.4)λlhl (p̂) = 0, ∀l ∈ V , (A.5)Rc (p̂c)− θ̂cCc (p̂c) ≤ 0, ∀c ∈ C, (A.6)hl (p̂) ≤ 0, ∀l ∈ V , (A.7)γ ≥ 0, α ≥ 0, λ ≥ 0, (A.8)(γ, α, λ)6= (0,0,0) . (A.9)132A.1. Proof of Proposition 2.1Next, we show by contraposition that γ 6= 0 necessarily holds. Suppose that γ = 0.Hence, we obtain from conditions (A.3) that αc = 0, ∀c ∈ U since Cc (p̂c) > 0, ∀c ∈ C.Due to the complementary slackness conditions in (A.5), we also have λl = 0 if l /∈ Va.Here, Va = {l ∈ V | hl (p̂) = 0} is the subset of primary users for which the interferenceconstraints are active. Thus, (A.2) implies:∑l∈Vaλl∇phl (p̂) = 0. (A.10)Furthermore, if γ = 0 and αc = 0, ∀c ∈ C, then the condition (A.9) implies that∑l∈Vaλl > 0. (A.11)Since the power constraint functions hl, ∀l ∈ V are affine, we can assume that∇phl (p) is apositive vector. Hence, we obtain from (A.11) that∑l∈Vaλl∇phl (p̂) > 0, which contradictsequation (A.10). Consequently, γ = 0 cannot hold.Since γ 6= 0, we can now redefine the parameters as ŵ = wγ and λ̂ =λγ and divideequations (A.2)-(A.5) by γ to obtain the equivalent equations (A.12)-(A.15):∑c∈Cα̂c(∇pRc (p̂c)− θ̂c∇pCc (p̂c))+L∑l=1λ̂l∇phl (p̂) = 0, (A.12)1− α̂cCc (p̂c) = 0, ∀c ∈ C, (A.13)α̂c(Rc (p̂c)− θ̂cCc (p̂c))= 0, ∀c ∈ C, (A.14)λ̂lhl (p̂) = 0, ∀l ∈ V , (A.15)Rc (p̂c)− θ̂cCc (p̂c) ≤ 0, ∀l ∈ V , (A.16)hl (p̂) ≤ 0, ∀l ∈ V , (A.17)α̂ ≥ 0, (A.18)λ̂ ≥ 0. (A.19)133A.2. Proof of Lemma 2.1Now, let us consider the parametric problem (Px) with a parameter x = (α, θ). ItsLagrangian is given by:Lx (p;λ) =∑c∈Cαc (Rc (pc)− θcCc (pc)) +∑∀l∈V,λlhl (p) . (A.20)We can identify equations (A.12), (A.15), (A.17) and (A.19) as the Karush-Kuhn-Tucker(KKT) optimality conditions [61] of (Px) with x =(α̂, θ̂). Since (Px) is convex, these KKTconditions are sufficient to prove that p̂ is also an optimal solution of (Px) [61].Finally, we obtain (2.14) from equation (A.13), i.e. p̂ satisfies:α̂cCc (p̂c)− 1 = 0, ∀c ∈ C. (A.21)By definition, we have Cc (pc) > 0, ∀c ∈ C, and with (A.21), we also have α̂c > 0, ∀c ∈C. Consequently, the complementary slackness conditions in (A.14) simplify to:Rc (p̂c)− θ̂cCc (p̂c) = 0, ∀c ∈ C. (A.22)Therefore, it is proved that an optimal solution of (P ) must satisfy the equations in (2.13)and (2.14).A.2 Proof of Lemma 2.1Here, we are going to prove the three properties of the matrix-valued function F (x) statedin Lemma 2.1. Recall that a function f : X → Y is said Lipschitz continuous on a set Xif and only if there exists a constant K ≥ 0 such that ‖f (x1)− f (x2)‖ ≤ K ‖x1 − x2‖ ,∀x1,x2 ∈ X .First, we prove that the Jacobian of F is given by (2.22). We can verify that the ob-jective function of the parametric problem (Px) is strictly convex by computing its secondderivative. Thus, its solution is unique for a given x, and we get from (2.17) and (2.18)134A.2. Proof of Lemma 2.1that:∂Fc(x)∂αd= 0, ∀d ∈ C, ∂Fc(x)∂θd = 0, ∀d ∈ C\ {c} ,∂Fc+|C|(x)∂θd= 0, ∀d ∈ C,∂Fc+|C|(x)∂αc= Cc (p˜c (x)) ,∂Fc(x)∂θc= Cc (p˜c (x)) ,∂Fc(x)∂αd= 0, ∀d ∈ C\ {c} .Correspondingly, the Jacobian matrix F′ (x) is given by the diagonal matrix in (2.22) andit is positive definite since Cc (pc) > 0, ∀c ∈ C by definition.Next, we prove the Lipschitz continuity and strong monotonicity of the matrix-valuedfunction F. By the definition in (2.17) and (2.18), F can be written as a composite functionG ◦ p˜ where G : R|U| → R2|U| is defined by the convex functions:Gc (p) = θcCc (p)− rc (p) , ∀c ∈ C, (A.23)Gc+|C| (p) = αcCc (p)− 1, ∀c ∈ C, (A.24)and p˜ (x) is a function that returns an optimal solution of the parametric convex problem(Px) for a given x.First, the component functions Gc and Gc+|C| are all Lipschitz continuous in the set Kdefined in (2.11) since K is compact and Gc and Gc+|U| are convex in K for each c ∈ C(see Theorem C.4.1 in [110]). Therefore, the matrix-valued function G is also Lipschitzcontinuous in the set K. Furthermore, by invoking Theorem D.1 in [111], it can be shownthat the function p˜ (x) is Lipschitz continuous in the set X defined in (2.19) since theparametric problem (Px) admits a unique solution for each x by strict convexity and sincethe set X is compact and convex (see also Lemma 2.1 in [112]). If p˜ and G are Lipschitzcontinuous in X and K respectively, then it implies that F =G ◦ p˜ is also Lipschitz contin-uous in K with a certain constant M . This result is due to Theorem 12.6 in [113]. Finally,135A.2. Proof of Lemma 2.1to show that F is strongly monotone, we write:y>F′ (x)y =∑c∈C(y2c + y2c+|U|)Cc (p˜c (x)) ≥2|C|∑c=1y2cmc ≥ m ‖y‖2 , ∀y ∈ R2|C| (A.25)where mc , minp∈K Cc (p) = Pf,c for each c ∈ C and m = minc∈U Pf,c > 0. The lastinequality in (A.25) implies that the function F is strongly monotone with a constantm > 0.Finally, we use the established properties above to show that F′ is also Lipschitz con-tinuous and that its inverse F′−1 is bounded. As shown previously, the function p˜ (x) isLipschitz continuous in the compact and convex set X . Thus, there is a constant L ≥ 0such that ‖p˜ (x1)− p˜ (x2)‖ ≤ L ‖x1 − x2‖ for all (x1,x2) ∈ X × X , and we obtain:|Cc (p˜ (x1))− Cc (p˜ (x2))| = ψc |p˜c (x1)− p˜c (x2)| ≤ ψcL ‖x1 − x2‖ , ∀c ∈ C. (A.26)From (2.22) and (A.26), we then have:‖F′ (x1)− F′ (x2)‖ ≤√2 |C|∑c∈C ψ2cL ‖x1 − x2‖ , (A.27)which means that F′ is also Lipschitz continuous in X . Moreover, its inverse F′−1 equals:F′−1 (x) =diag((1Cc(p˜(x)))c∈U)0|U|×|U|0|U|×|U| diag((1Cc(p˜(x)))c∈U). (A.28)Given the bounds of the setX in (2.20) and (2.21), each element 1Cc(p˜(x)) of F′−1 is boundedfrom above by αmaxc . Hence, there exists a constant B > 0 such that ‖F′−1 (x)‖ ≤ B.136A.3. Proof of Proposition 2.2A.3 Proof of Proposition 2.2Here, we prove in two steps that the system of equations F (x) = 0 has a unique solution.Before doing that, let us define the mapping P (x) = ProjX (x− µF (x)) where ProjX de-notes an orthogonal projection onto the convex set X . In addition, we select the parameterµ ∈(0, 2mM2)where m and M are the constants associated with the strong monotonicity andLipschitz continuity of F as previously shown.The first step in proving Proposition 2.2 is to show that P is a contractive mapping in Xfor all µ ∈(0, 2mM2). In this case, P is guaranteed to have a unique fixed point. The secondstep is then to prove that F (x) = 0 is equivalent to the fixed-point equation P (x) = x forall µ ∈(0, 2mM2). As a result, the solution of F (x) = 0 will also be unique.Proof that P (x) is a contractionLet us prove that P is a contraction. From the properties of F in Lemma 2.1, we can write:‖P (x1)− P (x2)‖2 = ‖ProjX (x1 − µF (x1))− ProjX (x2 − µF (x))‖2 (A.29)≤ ‖(x1 − µF (x1))− (x2 − µF (x))‖2 (A.30)= ‖x1 − x2‖2 − 2µ (x1 − x2)> (F (x1)− F (x2)) + µ2 ‖F (x1)− F (x2)‖2(A.31)≤ ‖x1 − x2‖2 − 2µm ‖x1 − x2‖2 + µ2 ‖F (x1)− F (x2)‖ (A.32)=(1− 2µm+ µ2M2)‖x1 − x2‖2 , (A.33)where the inequality (A.30) results from the non-expansiveness of the projection opera-tor [114]. The inequalities (A.32) and (A.33) are respectively obtained from the strongmonotonicity and Lipschitz continuity of F according to Lemma 2.1.From (A.33), we obtain ‖P (x1)− P (x2)‖ ≤√1− 2µm+ µ2M2 ‖x1 − x2‖. Since µ ∈137A.3. Proof of Proposition 2.2(0, 2mM2), we have√1− 2µm+ µ2M2 < 1. Thus, we showed that P is a contractive map-ping.Proof of equivalence between P (x) = x and F (x) = 0First, suppose that x̂ ∈ X satisfies F (x̂) = 0. Then, we have:P (x̂) = ProjX (x̂− µF (x̂)) = ProjX (x̂) = x̂which means that x̂ is a fixed point of P (x).Conversely, suppose that x̂ is the unique fixed point of the contractive mapping P for agiven µ ∈(0, 2mM2). In other words, we have P (x̂) = ProjX (x̂− µF (x̂)) = x̂. Then, we nextshow that x̂ also satisfies F (x̂) = 0. Since we assume that the problem (P ) is feasible, weknow that there exists x˜ ∈ X such that F (x˜) = 0. We then need to prove that x̂ = x˜. Forthat, we use the projection theorem (see Proposition 1.1.9 in [114]) which states that:(ProjX (z)− z)> (ProjX (z)− y) ≤ 0, ∀y ∈ X .Setting z := x̂− µF (x̂), we have:(ProjX (x̂− µF (x̂))− (x̂− µF (x̂)))> (ProjX (x̂− µF (x̂))− y) ≤ 0,which simplifies to F (x̂)> (x̂− y) ≤ 0, ∀y ∈ X since µ > 0. Next, using the fact thatF (x˜) = 0 and replacing y := x˜, we obtain:(F (x̂)− F (x˜))> (x̂− x˜) ≤ 0. (A.34)According to Lemma 2.1, F is strongly monotone with a constant m > 0. It means that:(F (x̂)− F (x˜))> (x̂− x˜) ≥ m ‖x̂− x˜‖2 . (A.35)Finally, we combine (A.34) and (A.35) into:138A.3. Proof of Proposition 2.20 ≤ (F (x˜)− F (x̂))> (x̂− x˜) ≤ −m ‖x̂− x˜‖2 , (A.36)which implies that x̂ = x˜ since m > 0.In summary, we showed that the system F (x) = 0 is equivalent to the fixed pointequation P (x) = x for any µ ∈(0, 2mM2). Furthermore, the solution of F (x) = 0 is uniquebecause the P (x) is a contraction and has a unique fixed point.139Appendix BProof of Result in Chapter 3Proof of Proposition 3.1First, we prove by induction that if we start with a feasible point of (P ) then the iteratesgenerated by the step (S.3) of Algorithm 3.1 are all feasible. Note that for any powerallocation p that satisfies the rate constraints c1 and the transmit power constraints c4, wecan always find a set of feasible power usage vectors{pg,ph,w}that satisfies the otherconstraints c2,3,5,6. Thus, we need only to prove that the iterates{p(n)}of the powerallocation are feasible. To do that, let us assume that p(n) is feasible for (P ), i.e for eachc ∈ M∪ K, we have Rc(p(n))≥ ρcT and∑u∈Ucpu,t ≤ Pc, ∀t = 1, . . . , T . Due to Lemma 3.1,we obtain:Rc(p(n))= R˜c(p(n);p(n))≥ ρcT, ∀c ∈M∪K. (B.1)(B.1) means that p(n) is also feasible for the subproblem(Qp(n)), which further impliesthat(Qp(n))has a non-empty feasible set.Then, step (S.2) of the algorithm gives us an optimal solution p(n+1) of subproblem(Qp(n)). This solution must also be feasible to(Qp(n))and satisfies:R˜c(p(n+1);p(n))≥ ρcT, ∀c ∈M∪K. (B.2)According to Lemma 3.1, the following inequality must also hold:Rc(p(n+1))≥ R˜c(p(n+1);p(n)), ∀c ∈M∪K (B.3)140Proof of Proposition 3.1since R˜c minorizes Rc. Combining (B.2) and (B.3), we obtain Rc(p(n+1))≥ ρcT , i.e. thenew iterate p(n+1) is feasible for (P ). By induction, if the initial power allocation p(0) isfeasible, then all subsequent iterates must remain feasible.Next, we prove that the proposed algorithm is a monotonically convergent and ascentalgorithm. In other words, we will show that:∑c∈M∪KEE(n+1)c ≥∑c∈M∪KEE(n)c , ∀n ∈ Nwhere EE(n+1)c is the energy efficiency of cell c achieved with the optimal solution p(n+1)of subproblem(Qp(n)), whose parameter is q = p(n). Using Lemma 1 again, we have theminorization relationship:∑c∈M∪KEE(n+1)c ≥∑c∈M∪KE˜Ec(p(n+1),pg (n+1)c ;p(n)). (B.4)Moreover, p(n+1) maximizes the objective function of subproblem(Qp(n)). Hence, we have:∑c∈M∪KE˜Ec(p(n+1),pg (n+1)c ;p(n))≥∑c∈M∪KE˜Ec(p(n),pg (n)c ;p(n)). (B.5)The right-hand side expression of (B.3) is equal to∑c∈M∪K EE(n)c according to Lemma1. Consequently, the last two inequalities (B.2) and (B.3) imply that∑c∈M∪K EE(n+1)c ≥∑c∈M∪K EE(n)c .141Appendix CProof of Results in Chapter 4C.1 Proof of Lemma 4.1After fixing the clustering variables to(xk,yk), the objective function becomes a convexfunction of only the beamforming variable v. Next, we reformulate the non-convex SINRconstraints in conic form [115]. Note that with simple manipulations, the SINR constraintsin (4.19) can be rewritten as:(1 +1ρu)∣∣hHu vu∣∣2 ≥∥∥∥∥∥∥hHu V σu∥∥∥∥∥∥2, ∀u (C.1)Moreover, any phase rotation on a beamforming vector vu preserves the feasibility and theobjective function value. It means we can multiply an optimal v∗u by an appropriate ejθuand obtain another feasible and optimal beamformer vˆ∗u = ejθuvu. In particular, we canchoose θu to make the inner product hHu vu real and positive for each user u. Since suchphase rotation does not change the problem, (C.1) is thus equivalent to:∥∥∥∥∥∥hHu V σu∥∥∥∥∥∥≤√1 +1ρuhHu vu, ∀u ∈ U (C.2)Re(hHu vu)≥ 0, ∀u ∈ U (C.3)Im(hHu vu)= 0, ∀u ∈ U (C.4)with (C.2) defining a second-order conic constraint [61]. As a result,(Sk)is convex. Sincethe second-order cone in (C.2) is closed and the power and interference limit constraints142C.2. Proof of Proposition 4.2define compact sets, their intersection, i.e. Fv, is bounded and closed, thus compact.C.2 Proof of Proposition 4.2In short, the equivalence between (M) and (P ′) is obtained from the dual representationsof the function g and of the feasible set H defined in (4.27)-(4.28). First, recall fromLemma 4.1 that the set Fv in (4.23) is convex, closed and bounded and that constraints(4.26) are convex for any fixed(xk,yk)∈ F(x,y). Therefore, we can apply Theorem 2.2 in[90] to express that (x,y) ∈ H if and only if it satisfies the infinite system:infv∈FvL (v, µ,x) ≤ 0, ∀µ ≥ 0 :∑(b,u)∈B×Uµbu = 1Since (S) is convex by Lemma 4.1, the strong duality theorem gives g (x,y) = supλ≥0infv∈FvL (v, λ,x,y) ,∀ (x,y) ∈ H ∩ F(x,y). Substituting g and H with their dual representations, problem (M)thus becomes:minimizex ysupλ≥0infv∈FvL (v, λ,x,y)subject to (x,y) ∈ F(x,y)0 ≥ infv∈FvL (v, µ,x) , ∀µ ≥ 0 :∑(b,u)∈B×U µbu = 1(C.5)Since Fv is convex and compact, the optimal solution of (S) is bounded. Thus, we canreplace the infimum in (C.5) with minimum. Using the definition of supremum as thesmallest upper-bound, we can obtain (4.32).143C.3. Proof of Lemma 4.2C.3 Proof of Lemma 4.2Since(vk, λk)is a pair of optimal solution and multiplier for(Sk), we have:vk = argminv∈FvL(v, λk,xk,yk)(C.6)= argminv∈Fv[f1 (v) + f2(xk,yk)+∑b∈B∑u∈Uλkbu(‖vbu‖22 − xkbu · Pmaxb)](C.7)= argminv∈Fv(f1 (v) +∑b∈B∑u∈Uλkbu ‖vbu‖22)(C.8)where (C.6) is due to the optimality of(vk, λk). After replacing the expression of L in(4.30) into (C.7) and removing the constant terms, we obtain the last equation (C.7).Now, we can rewrite the function ξ in (4.35) as:ξ(x,y;λk)= minv∈Fv[f1 (v) + f2 (x,y) +∑b∈B∑u∈Uλkbu(‖vbu‖22 − xbu · Pmaxb)](C.9)= f2 (x,y)−∑b∈B∑u∈UPmaxb λkbuxbu + minv∈Fv(f1 (v) +∑b∈B∑u∈Uλkbu ‖vbu‖22)(C.10)From (C.7), we see that the primal solution vk of(Sk)also solves the minimization termin (C.10). Thus, we have:ξ(x,y;λk)= f2 (x,y)−∑b∈B∑u∈UPmaxb λkbuxbu + f1(vk)+∑b∈B∑u∈Uλkbu∥∥vkbu∥∥22(C.11)After rearranging the terms in (C.11), we obtain the result ξ(x,y;λk)= L(vk, λk,x,y).C.4 Proof of Lemma 4.3First, we write the partial Langragian LF of the feasibility problem(F k)as:LF (v, α, µ, x) = α +∑b∈B∑u∈Uµbu(‖vbu‖22 − xbu · Pmaxb − α)(C.12)144C.4. Proof of Lemma 4.3Since(F k)is convex and(vk, αk)and µk are its optimal primal and dual solutions, thenwe have:(vk, αk)= argminv∈Fv, α≥0LF(v, α, µk, xk)(C.13)= argminv∈Fv, α≥0α(1−∑b∈B∑u∈Uµkbu)+∑b∈B∑u∈Uµkbu(‖vbu‖22 − xkbu · Pmaxb)(C.14)after rearranging the expression of LF in (C.12). With the optimality condition∂LF (v,α,µk,xk)∂α =0, we can show that µk must satisfy:∑b∈B∑u∈Uµkbu = 1 (C.15)By using the condition (C.15) in (C.14) and removing the constant terms in (C.14), weobtain:vk = argminv∈Fv∑b∈B∑u∈Uµkbu ‖vbu‖22 (C.16)Following the same procedure (C.9)-(C.11) as in the proof of Lemma 4.2, we can exploitthe linear separability of L with the fact (C.16) to show that ξ(x;µk)= L(vk, µk,x).145

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