Backward Penalty Schemes for Monotone Inclusion Problems

TL;DR

This paper introduces two iterative backward penalty schemes for solving complex monotone inclusion problems involving sums of set-valued and single-valued maximally monotone operators, with convergence depending on operator properties.

Contribution

It proposes novel backward penalty algorithms tailored for different types of single-valued operators in monotone inclusion problems, enhancing solution strategies for complex structures.

Findings

Two new iterative penalty schemes for monotone inclusions.

Schemes adapt to cocoercive and Lipschitz continuous operators.

Addresses complex structured problems with convergence guarantees.

Abstract

In this paper we are concerned with solving monotone inclusion problems expressed by the sum of a set-valued maximally monotone operator with a single-valued maximally monotone one and the normal cone to the nonempty set of zeros of another set-valued maximally monotone operator. Depending on the nature of the single-valued operator, we will propose two iterative penalty schemes, both addressing the set-valued operators via backward steps. The single-valued operator will be evaluated via a single forward step if it is cocoercive, and via two forward steps if it is monotone and Lipschitz continuous. The latter situation represents the starting point for dealing with complexly structured monotone inclusion problems from algorithmic point of view.