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Variational Analysis of Proximal Compositions and Integral Proximal Mixtures

Patrick L. Combettes, Diego J. Cornejo

TL;DR

The article develops a comprehensive variational analysis of two convexity-preserving constructions: proximal compositions $L\overset{\gamma}{\diamond}g$ and proximal cocompositions $L\overset{\gamma}{\blackdiamond}g$, along with their integral analogues via integral proximal mixtures and proximal expectations. It establishes a wide range of convex-analytic properties, including Legendre conjugacy, differentiability, Moreau envelopes, coercivity, and explicit proximal operators, and it studies how these constructions behave as the parameter $\gamma$ varies, including epi-convergence results. The work further extends to integral proximal mixtures and proximal expectations, showing that the proximal operators decompose naturally into integrals of the individual proximal operators and linking these concepts to direct Hilbert space integrals. Collectively, the results provide a robust theoretical framework for variational models that combine multiple convex functions and linear operators, with implications for efficient algorithm design in signal processing and related fields. The findings unify and extend finite-proximal mixtures to the integral setting, offering explicit formulas, duality relations, and asymptotic behavior that aid both analysis and computation.

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.

Variational Analysis of Proximal Compositions and Integral Proximal Mixtures

TL;DR

The article develops a comprehensive variational analysis of two convexity-preserving constructions: proximal compositions and proximal cocompositions , along with their integral analogues via integral proximal mixtures and proximal expectations. It establishes a wide range of convex-analytic properties, including Legendre conjugacy, differentiability, Moreau envelopes, coercivity, and explicit proximal operators, and it studies how these constructions behave as the parameter varies, including epi-convergence results. The work further extends to integral proximal mixtures and proximal expectations, showing that the proximal operators decompose naturally into integrals of the individual proximal operators and linking these concepts to direct Hilbert space integrals. Collectively, the results provide a robust theoretical framework for variational models that combine multiple convex functions and linear operators, with implications for efficient algorithm design in signal processing and related fields. The findings unify and extend finite-proximal mixtures to the integral setting, offering explicit formulas, duality relations, and asymptotic behavior that aid both analysis and computation.

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.
Paper Structure (11 sections, 39 theorems, 108 equations, 2 figures)

This paper contains 11 sections, 39 theorems, 108 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and background
  3. Proximal compositions
  4. General properties
  5. Convex-analytical properties
  6. Comparison with standard compositions and infimal postcompositions
  7. Asymptotic properties
  8. Integral proximal mixtures
  9. Definition and mathematical setting
  10. Properties
  11. Proximal expectations

Key Result

Lemma 2.1

Let $f$ and $g$ be functions from $\mathcal{H}$ to $\intv{{-}\infty}{{+}\infty}$. Then the following hold:

Figures (2)

  • Figure 1: Graphs of the proximal cocomposition and of the standard composition in Example \ref{['ex:1']}.
  • Figure 2: Graphs of the proximal cocomposition and of the standard composition in Example \ref{['ex:2']}.

Theorems & Definitions (97)

  • Definition 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 87 more