Asymmetric Forward-Backward-Adjoint Splitting for Solving Monotone Inclusions Involving Three Operators

TL;DR

This paper introduces a novel asymmetric splitting method for monotone inclusions involving three operators, unifying and extending existing algorithms, thereby broadening the scope of splitting techniques in convex optimization.

Contribution

It proposes the Asymmetric Forward-Backward-Adjoint splitting method, a new algorithm that generalizes and connects various primal-dual algorithms for structured convex problems.

Findings

Includes a Douglas-Rachford type scheme with a third cocoercive operator.

Unifies multiple existing splitting algorithms under a single framework.

Extends applicability of splitting methods to more complex problems.

Abstract

In this work we propose a new splitting technique, namely Asymmetric Forward-Backward-Adjoint splitting, for solving monotone inclusions involving three terms, a maximally monotone, a cocoercive and a bounded linear operator. Classical operator splitting methods, like Douglas-Rachford and Forward-Backward splitting are special cases of our new algorithm. Asymmetric Forward-Backward-Adjoint splitting unifies, extends and sheds light on the connections between many seemingly unrelated primal-dual algorithms for solving structured convex optimization problems proposed in recent years. More importantly, it greatly extends the scope and applicability of splitting techniques to a wider variety of problems. One important special case leads to a Douglas-Rachford type scheme that includes a third cocoercive operator.