Variational Analysis of Proximal Compositions and Integral Proximal Mixtures

TL;DR

This paper investigates the variational properties of new convexity-preserving constructs, namely proximal compositions and integral proximal mixtures, analyzing their convexity, conjugacy, differentiability, and asymptotic behavior.

Contribution

It provides a comprehensive variational analysis of the proximal composition and integral proximal mixture, extending understanding of their properties and applications in convex analysis.

Findings

Established convexity and conjugacy properties.

Analyzed differentiability and asymptotic behavior.

Discussed special case of proximal expectation.

Abstract

This paper establishes various variational properties of parametrized versions of two convexity-preserving constructs that were recently introduced in the literature: the proximal composition of a function and a linear operator, and the integral proximal mixture of arbitrary families of functions and linear operators. We study in particular convexity, Legendre conjugacy, differentiability, Moreau envelopes, coercivity, minimizers, recession functions, and perspective functions of these constructs, as well as their asymptotic behavior as the parameter varies. The special case of the proximal expectation of a family of functions is also discussed.