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Dissertation Defence: Strong Convergence of Recent Splitting Algorithms and Applications of Proximal Mappings
April 12 at 9:00 am - 1:00 pm
Shambhavi Singh, supervised by Dr. Heinz Bauschke, will defend their dissertation titled “Strong Convergence of Recent Splitting Algorithms and Applications of Proximal Mappings” in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics.
An abstract for Shambhavi Singh’s dissertation is included below.
Examinations are open to all members of the campus community as well as the general public. Please email heinz.bauschke@ubc.ca to receive the Zoom link for this exam.
ABSTRACT
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, then 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 operate in the smaller space X^{n-1}. 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.
