On the convergence of the forward-backward splitting method with linesearches

TL;DR

This paper analyzes the convergence of a forward-backward splitting method with new linesearches for nonsmooth convex optimization, establishing weak convergence without Lipschitz gradient assumptions and providing complexity results.

Contribution

Introduces two novel linesearches for the forward-backward method, proving weak convergence without requiring Lipschitz continuity of the gradient.

Findings

Weak convergence established under new linesearches

Convergence without Lipschitz gradient assumption

Complexity results for cost values with bounded stepsizes

Abstract

In this paper we focus on the convergence analysis of the forward-backward splitting method for solving nonsmooth optimization problems in Hilbert spaces when the objective function is the sum of two convex functions. Assuming that one of the functions is Fr\'echet differentiable and using two new linesearches, the weak convergence is established without any Lipschitz continuity assumption on the gradient. Furthermore, we obtain many complexity results of cost values at the iterates when the stepsizes are bounded below by a positive constant.