Monotone Operator Theory in Convex Optimization

TL;DR

This paper explores the deep connections between monotone operator theory and convex optimization, highlighting their roles in analysis and algorithms, and introducing new transformations and insights for solving convex problems.

Contribution

It provides a comprehensive review of monotone operators in convex optimization and introduces novel transformations and connections to enhance algorithmic approaches.

Findings

Subdifferentials are maximally monotone operators.

Proximity operators are shown as resolvents of monotone operators.

New transformations map proximity operators to proximity operators.

Abstract

Several aspects of the interplay between monotone operator theory and convex optimization are presented. The crucial role played by monotone operators in the analysis and the numerical solution of convex minimization problems is emphasized. We review the properties of subdifferentials as maximally monotone operators and, in tandem, investigate those of proximity operators as resolvents. In particular, we study new transformations which map proximity operators to proximity operators, and establish connections with self-dual classes of firmly nonexpansive operators. In addition, new insights and developments are proposed on the algorithmic front.