On the behaviour of algorithms featuring compositions of projectors and proximal mappings with no solutions
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: Alwadani, Salihah Thabet
issued: 2021-11-22T08:00:00Z
issued: 2021
related degree: Doctor of Philosophy - PhD
degree grantor: University of British Columbia
description: Finding a zero of the sum of two maximally monotone operators is of fundamental importance in variational analysis and optimization. There are various algorithms for solving this problem such as the Douglas–Rachford splitting algorithm (DR) and Arag ́on Artacho-Campoy algorithm (AACA), which is a new method in this area. Moreover, it is well known that there is a connection between the set of zeros of maximally monotone operators and the fixed point set of composition of nonexpansive operators. In this thesis we will explore the connection between algorithms and fixed point sets. This thesis consists of two parts. In the first part of this thesis, we advance the analysis of AACA. We use the proximal and resolvent average in our analysis. We re-derive the central convergence result on AACA and demonstrate that the underlying curve converges to the nearest zero of the sum of the two operators. Using the Attouch-Th ́era duality, we study the cycles, gap vectors and fixed point sets of compositions of general nonexpansive mappings in the second part. We also use the framework of monotone operator theory to resolve the geometry conjecture completely, which states that the fixed point set of the composition of projectors onto nonempty closed convex sets in Hilbert space is actually equal to the intersection of certain translations of the underlying sets. Then we extend the results from projectors to proximal mappings.
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/80257
note: On the behaviour of algorithmsfeaturing compositions of projectorsand proximal mappings with nosolutionsbySalihah Thabet AlwadaniB.Sc. Hons., Umm Al-Qura University, 2013M.Sc., Acadia University, 2017A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE COLLEGE OF GRADUATE STUDIES(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)November 2021c© Salihah Thabet Alwadani, 2021AbstractFinding a zero of the sum of two maximally monotone operators is offundamental importance in variational analysis and optimization. There arevarious algorithms for solving this problem such as the Douglas–Rachfordsplitting algorithm (DR) and Arago´n Artacho-Campoy algorithm (AACA),which is a new method in this area. Moreover, it is well known that thereis a connection between the set of zeros of maximally monotone operatorsand the fixed point set of composition of nonexpansive operators. In thisthesis we will explore the connection between algorithms and fixed pointsets. This thesis consists of two parts.In the first part of this thesis, we advance the analysis of AACA. Weuse the proximal and resolvent average in our analysis. We re-derive thecentral convergence result on AACA and demonstrate that the underlyingcurve converges to the nearest zero of the sum of the two operators.Using the Attouch-The´ra duality, we study the cycles, gap vectors andfixed point sets of compositions of general nonexpansive mappings in thesecond part. We also use the framework of monotone operator theory toresolve the geometry conjecture completely, which states that the fixed pointset of the composition of projectors onto nonempty closed convex sets inHilbert space is actually equal to the intersection of certain translations ofthe underlying sets. Then we extend the results from projectors to proximalmappings.iiLay SummaryOptimization and feasibility problems are important in mathematics andthe natural sciences. These are solved by algorithms featuring …
has type: Thesis/Dissertation
date issued: 2022-02
is shown at: 10.14288/1.0403824
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: Bauschke, Heinz H.
type: Text
identifier URI: http://hdl.handle.net/2429/80257