Four Operator Splitting via a Forward-Backward-Half-Forward Algorithm with Line Search

TL;DR

This paper introduces a novel splitting algorithm for monotone inclusions involving four operators, utilizing line search to adaptively determine step sizes, and demonstrates its effectiveness on constrained convex optimization problems.

Contribution

It generalizes existing splitting methods to handle four operators with a line search, improving efficiency in solving complex monotone inclusion problems.

Findings

Algorithm successfully solves non-linearly constrained least-squares problems.

Line search reduces the number of operator activations per iteration.

Performance comparisons show advantages over existing methods.

Abstract

In this article we provide a splitting method for solving monotone inclusions in a real Hilbert space involving four operators: a maximally monotone, a monotone-Lipschitzian, a cocoercive, and a monotone-continuous operator. The proposed method takes advantage of the intrinsic properties of each operator, generalizing the forward-back-half forward splitting and the Tseng's algorithm with line-search. At each iteration, our algorithm defines the step-size by using a line search in which the monotone-Lipschitzian and the cocoercive operators need only one activation. We also derive a method for solving non-linearly constrained composite convex optimization problems in real Hilbert spaces. Finally, we implement our algorithm in a non-linearly constrained least-square problem, and we compare its performance with available methods in the literature.