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Ouyang, Hui

University of British Columbia

This dissertation consists of two parts. Part I concerns circumcentered methods for best approximation or feasibility problems while Part II deals with generalized proximal point algorithms for monotone inclusion problems. The monotone inclusion problem has favourable modelling capability for a large number of applications in optimization. In particular, monotone inclusion problems cover the important class of best approximation and feasibility problems. The circumcentered method has great performance for solving some best approximation or feasibility problems although its iteration mapping, the circumcenter mapping, is in general not globally well-defined. One of the most popular algorithms for solving monotone inclusion problems is the generalized proximal point algorithm. In Part I, we investigate the best approximation mapping that is a generalized version of the circumcenter mapping, explore locally proper circumcenter mappings and circumcentered methods, and consider circumcenters in the framework of general Bregman distances. In Part II, we study the weak, strong, and linear convergence of generalized proximal point algorithms from the following three directions: study the stability of iteration algorithms involving nonexpansive operators; discuss the convergence of relaxation variants of the Krasnosel'skii-Mann iteration; directly work on the convergence of generalized proximal point algorithms by using the beautiful properties of the resolvent of maximally monotone operators.

Text

2022-07-22

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/82194

Mathematics

University of British Columbia

UBCO

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