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Alwadani, Salihah Thabet

University of British Columbia

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.

Text

2021-11-22

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/80257

Mathematics

University of British Columbia

UBCO

http://creativecommons.org/licenses/by-nc-nd/4.0/