Union Averaged Operators with Applications to Proximal Algorithms for Min-Convex Functions

TL;DR

This paper introduces union averaged nonexpansive operators, a new class of set-valued operators, and applies them to analyze proximal algorithms for minimizing sums of convex functions, including non-convex cases.

Contribution

It defines and studies the properties of union averaged nonexpansive operators and demonstrates their application in proximal algorithms for complex convex minimization problems.

Findings

Union averaged nonexpansive operators are closed under unions, convex combinations, and compositions.

Fixed point iterations of these operators are locally convergent near strong fixed points.

Applications include analyzing proximal algorithms for sums of convex and finitely-minimal convex functions.

Abstract

In this paper we introduce and study a class of structured set-valued operators which we call union averaged nonexpansive. At each point in their domain, the value of such an operator can be expressed as a finite union of single-valued averaged nonexpansive operators. We investigate various structural properties of the class and show, in particular, that is closed under taking unions, convex combinations, and compositions, and that their fixed point iterations are locally convergent around strong fixed points. We then systematically apply our results to analyze proximal algorithms in situations where union averaged nonexpansive operators naturally arise. In particular, we consider the problem of minimizing the sum two functions where the first is convex and the second can be expressed as the minimum of finitely many convex functions.