Resolvent of the parallel composition and proximity operator of the infimal postcomposition

TL;DR

This paper derives the resolvent of the parallel composition of maximally monotone operators and the proximity operator of infimal postcomposition, extending convex optimization tools and generalizing Moreau's decomposition.

Contribution

It provides explicit formulas for resolvents and proximity operators in composite convex optimization, under mild assumptions, extending existing theory.

Findings

Explicit resolvent computation for parallel composition

Proximity operator of infimal postcomposition derived

Generalized Moreau's decomposition introduced

Abstract

In this paper we provide the resolvent computation of the parallel composition of a maximally monotone operator by a linear operator under mild assumptions. Connections with a modification of the warped resolvent are provided. In the context of convex optimization, we obtain the proximity operator of the infimal postcomposition of a convex function by a linear operator and we extend full range conditions on the linear operator to mild qualification conditions. We also introduce a generalization of the proximity operator involving a general linear bounded operator leading to a generalization of Moreau's decomposition for composite convex optimization.