Convergence Analysis of a Proximal Point Algorithm for Minimizing Differences of Functions

TL;DR

This paper introduces a generalized proximal point algorithm for minimizing differences of functions, specifically targeting nonconvex optimization problems, and analyzes its convergence under the Kurdyka-Łojasiewicz property.

Contribution

It proposes a novel algorithm for nonconvex optimization involving differences of functions and provides convergence analysis under a broad mathematical condition.

Findings

Algorithm converges under Kurdyka-Łojasiewicz property

Extends proximal methods to nonconvex difference-of-functions problems

Provides theoretical convergence guarantees

Abstract

Several optimization schemes have been known for convex optimization problems. However, numerical algorithms for solving nonconvex optimization problems are still underdeveloped. A progress to go beyond convexity was made by considering the class of functions representable as differences of convex functions. In this paper, we introduce a generalized proximal point algorithm to minimize the difference of a nonconvex function and a convex function. We also study convergence results of this algorithm under the main assumption that the objective function satisfies the Kurdyka - \L ojasiewicz property.