Generalized solutions for the sum of two maximally monotone operators

TL;DR

This paper introduces a generalized 'normal problem' for finding solutions to the sum of two maximally monotone operators, extending solution concepts to cases where solutions may not exist, with strong variational and duality properties.

Contribution

It proposes a novel 'normal problem' framework that generalizes solutions for sums of maximally monotone operators, applicable even when original solutions do not exist.

Findings

Normal problem coincides with original solutions when they exist

Normal problem provides solutions in infeasible cases

Theoretical properties include variational and duality advantages

Abstract

A common theme in mathematics is to define generalized solutions to deal with problems that potentially do not have solutions. A classical example is the introduction of least squares solutions via the normal equations associated with a possibly infeasible system of linear equations. In this paper, we introduce a "normal problem" associated with finding a zero of the sum of two maximally monotone operators. If the original problem admits solutions, then the normal problem returns this same set of solutions. The normal problem may yield solutions when the original problem does not admit any; furthermore, it has attractive variational and duality properties. Several examples illustrate our theory.