Epiconvergence, the Moreau envelope and generalized linear-quadratic functions

TL;DR

This paper introduces generalized linear-quadratic functions built from monotone relations, develops their calculus and envelope properties, and explores their limits and characterizations within convex analysis.

Contribution

It generalizes existing results by defining a new class of functions and analyzing their properties and limits using Moreau envelopes and epigraphical convergence.

Findings

Characterization of when quadratic functions are Moreau envelopes of generalized linear-quadratic functions.

Development of calculus rules for this new class of functions.

Analysis of epigraphical limits of quadratic functions in finite-dimensional spaces.

Abstract

This work introduces the class of generalized linear-quadratic functions, constructed using maximally monotone symmetric linear relations. Calculus rules and properties of the Moreau envelope for this class of functions are developed. In finite dimensions, on a metric space defined by Moreau envelopes, we consider the epigraphical limit of a sequence of quadratic functions and categorize the results. We explore the question of when a quadratic function is a Moreau envelope of a generalized linear-quadratic function; characterizations involving nonexpansiveness and Lipschitz continuity are established. This work generalizes some results by Hiriart-Urruty and by Rockafellar and Wets.