Regularities in variational analysis and their applications in nonconvex optimization
department or school: Science, Irving K. Barber Faculty of (Okanagan)
department or school: Computer Science, Mathematics, Physics and Statistics, Department of (Okanagan)
data provider: DSpace
degree campus: UBCO
creator: Wang, Ziyuan
issued: 2025-06-03T14:58:43Z
issued: 2025
related degree: Doctor of Philosophy - PhD
degree grantor: University of British Columbia
description: In this thesis, we investigate regularities in variational analysis and explore their algorithmic applications. The term "regularities" refers to favorable properties of functions and operators that lead to pleasant behavior of algorithms designed for convex problems in the absence of convexity, including the Kurdyka-Åojasiewicz property and variational convexity. We extend the forward-reflected-backward splitting methodâa convex optimization algorithmâand its variations to nonconvex setting, both with and without a traditional assumption on Lipschitz smoothness. Our analysis hinges on analyzing merit functions associated with nonconvex extensions of the forward-reflected-backward splitting method that decrease along iterates generated by these algorithms. The Kurdyka-Åojasiewicz property guarantees these extensions converge globally to a stationary point of the objective function and more. We investigate systematically the level proximal subdifferential by characterizing its existence, single-valuedness, connection to Lipschitz smoothness, and more. These results reveal the level proximal subdifferential provides deep insights into variational analysis and optimization, which, among other things, lead to novel characterizations of variational convexity and of Lipschitz-type properties of proximal operators in the absence of convexity. We also investigate calculus rules and algorithmic applications of variational convexity. Aided by these results, new insights regarding the local behavior of several celebrated algorithms in nonconvex setting are obtained. Finally, we introduce the left and right Bregman level proximal subdifferentials and investigate them systematically. New correspondences among properties of the Bregman proximal operators, the underlying functions, and the associated subdifferential operators are also established. Asymmetry and duality gap occur in this process as consequences of Bregman distance.
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/91263
note: Regularities in Variational Analysisand Their Applications in NonconvexOptimizationbyZiyuan WangB.S., Chongqing Jiaotong Univeristy, 2018M.Sc., The University of British Columbia, 2020A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinIrving K. Barber Faculty of Science(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)May 2025© Ziyuan Wang, 2025The following individuals certify that they have read, and recommendto the College of Graduate Studies for acceptance, a thesis/dissertation en-titled:Regularities in Variational Analysis and Their Applicationsin Nonconvex Optimizationsubmitted by Ziyuan Wang in partial fulfilment of the requirements of thedegree of Doctor of PhilosophyDr. Xianfu Wang, Irving K. Barber Faculty of ScienceSupervisorDr. Heinz Bauschke, Irving K. Barber Faculty of ScienceSupervisory Committee MemberDr. John Braun, Irving K. Barber Faculty of ScienceSupervisory Committee MemberDr. Jahangir Hossain, School of EngineeringUniversity ExaminerDr. James Burke, Univeristy of WashingtonExternal ExamineriiAbstractIn this thesis, we investigate regularities in variational analysis and exploretheir algorithmic applications. The term âregularitiesâ refers to favorableproperties of functions and operators that lead to pleasant behavior of algo-rithms designed for convex problems in the absence of convexity, includingthe Kurdyka- Lojasiewicz property and variational convexity.We extend the forward-reflected-backward splitting methodâa convexoptimization algorithmâand its variations to nonconvex setting, both withand without a traditional assumption on Lipschitz smoothness. Our analysishinges on analyzing merit functions associated with nonconvex extensions ofthe forward-reflected-backward splitting method that decrease along iteratesgenerated by these algorithms. The Kurdyka- Lojasiewicz property guaran-tees these extensions converge globally to a stationary point of the objectivefunction, as well as prov…
has type: Thesis/Dissertation
date issued: 2025-09
is shown at: 10.14288/1.0449028
language: eng
degree discipline: Mathematics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivatives 4.0 International
rights URI: http://creativecommons.org/licenses/by-nc-nd/4.0/
scholar level: Graduate
contributor: Wang, Xianfu
type: Text
identifier URI: http://hdl.handle.net/2429/91263