Auxiliary problem principle and inexact variable metric forward-backward algorithm for minimizing the sum of a differentiable function and a convex function

TL;DR

This paper introduces inexact auxiliary problem and variable metric forward-backward algorithms for minimizing the sum of a differentiable and a convex function, proving convergence under weaker conditions than previous methods.

Contribution

It develops new descent algorithms based on inexact auxiliary problems and variable metrics, with convergence guarantees under the Kurdyka-Lojasiewicz inequality.

Findings

Proposed algorithms converge under weaker assumptions.

Algorithms effectively handle composite optimization problems.

Convergence proofs extend existing theoretical frameworks.

Abstract

In view of the minimization of a function which is the sum of a differentiable function $f$ and a convex function $g$ we introduce descent methods which can be viewed as produced by inexact auxiliary problem principleor inexact variable metric forward-backward algorithm. Assuming that the global objective function satisfies the Kurdyka-Lojasiewicz inequalitywe prove the convergence of the proposed algorithm weakening assumptions found in previous works.