Convergence analysis of a variable metric forward-backward splitting algorithm with applications

TL;DR

This paper introduces a new convergence analysis for a variable metric forward-backward splitting algorithm with extended relaxation parameters, applicable to various convex optimization problems and variational inequalities, with demonstrated effectiveness in LASSO.

Contribution

It provides the first weak convergence proof for this class of variable metric algorithms with extended relaxation, broadening their theoretical foundation and practical applicability.

Findings

Proves weak convergence under weak conditions on relaxation parameters.

Develops a variable metric algorithm for convex minimization with Lipschitz gradient.

Shows effectiveness through numerical results on LASSO problem.

Abstract

The forward-backward splitting algorithm is a popular operator-splitting method for solving monotone inclusion of the sum of a maximal monotone operator and a cocoercive operator. In this paper, we present a new convergence analysis of a variable metric forward-backward splitting algorithm with extended relaxation parameters in real Hilbert spaces. We prove that this algorithm is weakly convergent when certain weak conditions are imposed upon the relaxation parameters. Consequently, we recover the forward-backward splitting algorithm with variable step sizes. As an application, we obtain a variable metric forward-backward splitting algorithm for solving the minimization problem of the sum of two convex functions, where one of them is differentiable with a Lipschitz continuous gradient. Furthermore, we discuss the applications of this algorithm to the fundamental of the variational inequalities problem, constrained convex minimization problem, and split feasibility problem. Numerical experimental results on LASSO problem in statistical learning demonstrate the effectiveness of the proposed iterative algorithm.