Convergence of Descent Optimization Algorithms under Polyak-Łojasiewicz-Kurdyka Conditions

Glaydston Bento (), Boris Mordukhovich (), Tiago Mota () and Yurii Nesterov ()
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Glaydston Bento: Federal University of Goiás
Boris Mordukhovich: Wayne State University
Tiago Mota: Federal University of Goiás
Yurii Nesterov: Corvinus University of Budapest and School of Data Sciences (Chinese University of Hong Kong (Shenzhen); part time).

Journal of Optimization Theory and Applications, 2025, vol. 207, issue 3, No 1, 29 pages

Abstract: Abstract This paper develops a comprehensive convergence analysis for generic classes of descent algorithms in nonsmooth and nonconvex optimization under several conditions of the Polyak-Łojasiewicz-Kurdyka (PLK) type. Along other results, we prove the finite termination of generic algorithms under the PLK conditions with lower exponents. Specifications are given to establish new convergence rates for inexact reduced gradient methods and some versions of the boosted algorithm in DC programming. It is revealed, e.g., that the lower exponent PLK conditions for a broad class of difference programs are 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.

Keywords: Nonsmooth optimization; Descent methods; Global convergence analysis; Polyak-Łojasiewicz-Kurdyka conditions.; Primary 65K05; 65K10; 90C26; 47N10 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10957-025-02816-z

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