Solving Composite Monotone Inclusions in Reflexive Banach Spaces by Constructing Best Bregman Approximations from Their Kuhn-Tucker Set

TL;DR

This paper introduces a novel operator splitting method for solving composite monotone inclusions in reflexive Banach spaces, utilizing Bregman approximations and achieving strong convergence without extra restrictions.

Contribution

It is the first to develop an operator splitting approach outside Hilbert spaces that constructs best Bregman approximations from the Kuhn-Tucker set.

Findings

Method achieves strong convergence in reflexive Banach spaces.

It operates without restrictions on monotone operators or norms.

Provides an alternative to proximal methods even in Euclidean spaces.

Abstract

We introduce the first operator splitting method for composite monotone inclusions outside of Hilbert spaces. The proposed primal-dual method constructs iteratively the best Bregman approximation to an arbitrary point from the Kuhn-Tucker set of a composite monotone inclusion. Strong convergence is established in reflexive Banach spaces without requiring additional restrictions on the monotone operators or knowledge of the norms of the linear operators involved in the model. The monotone operators are activated via Bregman distance-based resolvent operators. The method is novel even in Euclidean spaces, where it provides an alternative to the usual proximal methods based on the standard distance.