Dissertation Defence: Regularities in Variational Analysis and Their Applications in Nonconvex Optimization
May 9 at 9:30 am - 1:30 pm
Ziyuan Wang, supervised by Dr. Shawn Wang, will defend their dissertation titled “Regularities in Variational Analysis and Their Applications in Nonconvex Optimization” in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics.
An abstract for Ziyuan Wang’s dissertation is included below.
Examinations are open to all members of the campus community as well as the general public. Please email shawn.wang@ubc.ca to receive the Zoom link for this defence.
Abstract
In this thesis, we investigate regularities in variational analysis and explore their algorithmic applications.
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- Lojasiewicz property guarantees these extensions converge globally to a stationary point of the objective function, as well as provable convergence rates of these algorithms.
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. In particular, variational sufficiency, a special case of variational convexity, guarantees that proximal gradient method converges to local minimizers rather than just stationary points.
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.
