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On the Moreau envelope properties of weakly convex functions

Marien Renaud, Arthur Leclaire, Nicolas Papadakis

TL;DR

This work analyzes the Moreau envelope $g^{\gamma}$ of $\rho$-weakly convex functions, establishing its differentiability, dual relationships via convex conjugates, and preservation of minimizers and critical points, while detailing how convexity and strong convexity constants degrade or improve under regularization. It presents explicit formulas for $g^{\gamma}$, $\nabla g^{\gamma}$, and higher-order structure, and connects proximal operators to convex-conjugate duality and Hamilton–Jacobi dynamics. The results show that $g^{\gamma}$ remains a useful, smooth surrogate for optimization, with a quantified impact on convexity and a demonstration that the image of the proximal operator is almost convex. These insights support stable proximal-based methods and smoothing schemes for nonconvex, weakly convex objectives in optimization and inverse problems.

Abstract

In this document, we present the main properties satisfied by the Moreau envelope of weakly convex functions. The Moreau envelope has been introduced in convex optimization to regularize convex functionals while preserving their global minimizers. However, the Moreau envelope is also defined for the more general class of weakly convex function and can be a useful tool for optimization in this context. The main properties of the Moreau envelope have been demonstrated for convex functions and are generalized to weakly convex function in various works. This document summarizes the vast literature on the properties of the Moreau envelope and provides the associated proofs.

On the Moreau envelope properties of weakly convex functions

TL;DR

This work analyzes the Moreau envelope of -weakly convex functions, establishing its differentiability, dual relationships via convex conjugates, and preservation of minimizers and critical points, while detailing how convexity and strong convexity constants degrade or improve under regularization. It presents explicit formulas for , , and higher-order structure, and connects proximal operators to convex-conjugate duality and Hamilton–Jacobi dynamics. The results show that remains a useful, smooth surrogate for optimization, with a quantified impact on convexity and a demonstration that the image of the proximal operator is almost convex. These insights support stable proximal-based methods and smoothing schemes for nonconvex, weakly convex objectives in optimization and inverse problems.

Abstract

In this document, we present the main properties satisfied by the Moreau envelope of weakly convex functions. The Moreau envelope has been introduced in convex optimization to regularize convex functionals while preserving their global minimizers. However, the Moreau envelope is also defined for the more general class of weakly convex function and can be a useful tool for optimization in this context. The main properties of the Moreau envelope have been demonstrated for convex functions and are generalized to weakly convex function in various works. This document summarizes the vast literature on the properties of the Moreau envelope and provides the associated proofs.

Paper Structure

This paper contains 11 sections, 24 theorems, 123 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Overview of the document
  3. Definitions and first properties
  4. On the dependency of g gamma w.r.t. gamma positive
  5. The intrinsic link between the Moreau envelope and the convex conjugate
  6. On the preservation of the convexity by the Moreau envelope
  7. Properties of the proximal operator for weakly convex functions
  8. The Moreau envelope is differentiable and preserves the minimizers and the critical points
  9. Second derivative of the Moreau envelope and convexity of the image of the proximal operator
  10. Numerical examples
  11. Conclusion

Key Result

Lemma 3.1

If $g$ is $\rho$-weakly convex function with $\gamma \rho < 1$, for $x, y \in {\mathbb{R}}^d$ and $p = \textsf{Prox}_{\gamma g}(x)$, we have

Figures (5)

  • Figure 1: Graph of the function $h$, $h^\gamma$ for $\gamma \in \{0.1, 0.5, 0.99\}$ and their associated proximal operators.
  • Figure 2: Graph of the function $f$, $f^\gamma$ for $\gamma \in \{0.1, 0.25, 0.49\}$ and their associated proximal operators.
  • Figure 3: Graph of the function $u$, $u^\gamma$ for $\gamma \in \{0.1, 0.5, 0.8\}$ and their associated proximal operators.
  • Figure 4: Graph of the function $v$, $v^\gamma$ for $\gamma \in \{0.1, 0.5, 0.99\}$ and associated proximal operators.
  • Figure 5: Graph of the function $\psi$, $\psi^\gamma$ for $\gamma \in \{0.1, 0.5, 0.99\}$ and associated proximal operators.

Theorems & Definitions (60)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Lemma 3.1
  • Proof 1
  • Proposition 3.2
  • Proof 2
  • Proposition 3.3
  • Proof 3
  • ...and 50 more