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Wang, Ziyuan

University of British Columbia

In this thesis, we investigate regularities in variational analysis and explore their algorithmic applications. The term "regularities" refers to favorable properties of functions and operators that lead to pleasant behavior of algorithms designed for convex problems in the absence of convexity, including the Kurdyka-Łojasiewicz property and variational convexity. We extend the forward-reflected-backward splitting method—a convex optimization algorithm—and its variations to nonconvex setting, both with and without a traditional assumption on Lipschitz smoothness. Our analysis hinges on analyzing merit functions associated with nonconvex extensions of the forward-reflected-backward splitting method that decrease along iterates generated by these algorithms. The Kurdyka-Łojasiewicz property guarantees these extensions converge globally to a stationary point of the objective function and more. We investigate systematically the level proximal subdifferential by characterizing its existence, single-valuedness, connection to Lipschitz smoothness, and more. These results reveal the level proximal subdifferential provides deep insights into variational analysis and optimization, which, among other things, lead to novel characterizations of variational convexity and of Lipschitz-type properties of proximal operators in the absence of convexity. We also investigate calculus rules and algorithmic applications of variational convexity. Aided by these results, new insights regarding the local behavior of several celebrated algorithms in nonconvex setting are obtained. Finally, we introduce the left and right Bregman level proximal subdifferentials and investigate them systematically. New correspondences among properties of the Bregman proximal operators, the underlying functions, and the associated subdifferential operators are also established. Asymmetry and duality gap occur in this process as consequences of Bregman distance.

Text

2025-06-03

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/91263

Mathematics

University of British Columbia

UBCO

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