Convergence of Descent Methods under Polyak- Lojasiewicz-Kurdyka Properties
This talk presents a novel convergence analysis for a class of descent methods in nonsmooth and nonconvex optimization under the Polyak-Lojasiewicz-Kurdyka properties, proving finite termination under certain conditions and revealing incompatibilities between these properties and gradient Lipschitz continuity in certain frameworks.
Overview
This talk develops the novel convergence analysis of a generic class of descent methods in nonsmooth and nonconvex optimization under several versions of the Polyak- Lojasiewicz -Kurdyka (PLK) properties. Along with other results, we prove the finite termination of generic algorithms under the PL K property with lower exponents. Specifications are given to convergence rates of some particular algorithms including inexact reduced gradient methods and the boosted algorithm in DC programming. It is revealed, e.g., that the lower exponent PLK property in the DC framework is incompatible with the gradient Lipschitz continuity for the plus function around a local minimizer. On the other hand, we show that the above inconsistency observation may fail if the Lipschitz continuity is replaced by merely the gradient continuity. Based on joint work with G. Bento, T. Mota and Yu. Nesterov.
Presenters
Boris Mordukhovich, Distinguished Professor, Mathematics, Wayne State University, USA
Brief Biography
Boris Mordukhovich is an American mathematician, who was born in the Soviet Union and emigrated to the U.S.A. in 1988. Currently he is Distinguished University Professor and Lifetime Scholar of the Academy of Scholars at Wayne State University (Vice President, 2009–2010 and President, 2010–2011).
Boris is recognized for his research in the areas of nonlinear analysis, optimization, and control theory, and is one of the founders of modern variational analysis. He developed constructions of generalized differentiation, bearing now his name. His theory and its applications in many fields have been summarized in a 2-volume monograph. He has published more than 500 journal papers and several books.
Boris is an AMS Fellow of the Inaugural Class, a SIAM Fellow, and a recipient of many international awards and honors, including Dr. Honoris Causa from the National Sun Yat-sen University in Taiwan, University of Messina, University of Alicante in Spain, Babeș-Bolyai University, Romania, and from Vietnam Academy of Science and Technology. He is the Founding Editor (2008) and was a co-Editor-in-Chief (2009–2014) of the journal Set-Valued and Variational Analysis. Since 2016, he is a co-Editor-in-Chief of Applied Analysis and Optimization, and since 2021 he is the area editor of Journal of Optimization Theory and Application. He was Chair (2012-2015) of the International Working Group on Generalized Convexity and Monotonicity. In 2016, he was elected to the Accademia Peloritana dei Pericolanti in Italy, and in 2021 he became a Foreign Member of the National Academy of Sciences of Ukraine. He is on the list of Highly Cited Researchers in Mathematics and Inaugural ScholarGPS Highly Ranked Scholar in Mathematical Optimization, 2024.
