The resolvent order: a unification of the orders by Zarantonello, by Loewner, and by Moreau

TL;DR

This paper introduces the resolvent order, unifying existing orders for matrices, projectors, and convex functions, and explores its properties and applications in convex analysis.

Contribution

It defines the resolvent order, connecting and unifying several well-known orders in matrix analysis and convex optimization, and constructs related partial orders.

Findings

The resolvent order unifies Loewner and Zarantonello orders.

Connections between the resolvent order and Moreau's order are established.

Examples illustrate the applicability of the new order in convex analysis.

Abstract

We introduce and investigate the resolvent order, which is a binary relation on the set of firmly nonexpansive mappings. It unifies well-known orders introduced by Loewner (for positive semidefinite matrices) and by Zarantonello (for projectors onto convex cones). A connection with Moreau's order of convex functions is also presented. We also construct partial orders on (quotient sets of) proximal mappings and convex functions. Various examples illustrate our results.