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Singh, Shambhavi

University of British Columbia

The problem of finding a zero of the sum of n maximally monotone operators is of central importance in the field of optimization. When these operators are subdifferentials, one encounters the problem of finding a minimizer of the sum of n functions. When n=2, the celebrated Douglas-Rachford splitting algorithm generates a sequence in the underlying space X that is known to converge weakly to a zero of the sum of the operators. When n ≥ 3, one can solve the general sum problem by working in X^n. Recently, several new algorithms have been proposed by Ryu, by Malitsky and Tam and by Campoy. These algorithms operate in X^(n−1), which is smaller than X^n. Relatedly, Bredies, Chenchene, Lorenz and Naldi present a framework which is able to encapsulate the Douglas-Rachford, the Chambolle-Pock and other algorithms. In this thesis, we explore these new algorithms for normal cone operators of closed linear subspaces. This allows us to strengthen existing convergence results. We also explore resolvents and splitting methods. First, we improve Carlier’s recent refinement of the Fenchel-Young inequality with respect to duality and cyclic monotonicity. Secondly, we present a formula for finding matrices with provided (scaled) row and column sums given a starting point based on a new projection formula. Finally, inspired by recent work by Aharoni, Censor and Jiang, we propose new algorithms for finding best approximation pairs that perform well in our numerical experiments.

Text

2024-04-29

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/88100

Mathematics

University of British Columbia

UBCO

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