Convergence of Descent Optimization Algorithms under Polyak-\L ojasiewicz-Kurdyka Conditions

TL;DR

This paper provides a comprehensive convergence analysis for descent algorithms in nonsmooth, nonconvex optimization under Polyak-ojasiewicz-Kurdyka conditions, including finite termination and new convergence rates.

Contribution

It introduces new convergence results and rates for descent algorithms under PLK conditions, especially for inexact methods and difference programming, expanding theoretical understanding.

Findings

Finite termination under lower exponents of PLK conditions.

New convergence rates for inexact reduced gradient methods.

Incompatibility of PLK conditions with Lipschitz continuity in certain cases.

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-\L ojasiewicz-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.