Resolvent and Proximal Compositions

TL;DR

This paper introduces the resolvent and proximal compositions, new operations that preserve monotonicity and convexity respectively, linking them to existing concepts and applying them to equilibrium and optimization problems.

Contribution

The paper defines the resolvent and proximal compositions, explores their properties, and demonstrates their applications in monotone inclusion and convex optimization.

Findings

Resolvant and proximal compositions preserve key properties like monotonicity and convexity.

The compositions unify and extend known concepts such as resolvent and proximal averages.

Applications include relaxation techniques for monotone inclusion and convex optimization problems.

Abstract

We introduce the resolvent composition, a monotonicity-preserving operation between a linear operator and a set-valued operator, as well as the proximal composition, a convexity-preserving operation between a linear operator and a function. The two operations are linked by the fact that, under mild assumptions, the subdifferential of the proximal composition of a convex function is the resolvent composition of its subdifferential. The resolvent and proximal compositions are shown to encapsulate known concepts, such as the resolvent and proximal averages, as well as new operations pertinent to the analysis of equilibrium problems. A large core of properties of these compositions is established and several instantiations are discussed. Applications to the relaxation of monotone inclusion and convex optimization problems are presented.