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New Characterizations of Nonsmooth Convex Functions via Generalized Derivatives

Vo Thanh Phat

Abstract

This paper studies the convexity properties of nonsmooth extended-real-valued weakly convex functions, a class of functions that is central to modern optimization and its applications. We establish new characterizations of convexity using second-order generalized derivative tools, including subgradient graphical derivatives, second subderivatives, and second-order subdifferentials. These tools allow us to derive necessary and sufficient conditions for convexity in the nonsmooth framework.

New Characterizations of Nonsmooth Convex Functions via Generalized Derivatives

Abstract

This paper studies the convexity properties of nonsmooth extended-real-valued weakly convex functions, a class of functions that is central to modern optimization and its applications. We establish new characterizations of convexity using second-order generalized derivative tools, including subgradient graphical derivatives, second subderivatives, and second-order subdifferentials. These tools allow us to derive necessary and sufficient conditions for convexity in the nonsmooth framework.

Paper Structure

This paper contains 6 sections, 11 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries from Variational Analysis
  3. Characterizations via Subgradient Graphical Derivatives
  4. Characterizations via Second Subderivatives
  5. Characterizations via Second-Order Subdifferentials
  6. Conclusion

Key Result

Proposition 2.1

Let $\varphi\colon{\rm I\!R}^n\to\overline{{\rm I\!R}}$ be an l.s.c. and prox-bounded function with threshold $\lambda_\varphi >0$. Suppose that $\varphi$ is continuously prox-regular at $\bar{x}$ for $\bar{v}\in\partial\varphi(\bar{x})$, then for all sufficiently small numbers $\lambda \in (0,\lamb In the case where $\varphi$ is weakly convex, then $\nabla e_\lambda\varphi$ is globally Lipschitz

Theorems & Definitions (13)

  • Proposition 2.1: Moreau envelopes of prox-regular functions
  • Theorem 3.1: necessary condition of convex functions via subgradient graphical derivatives
  • Lemma 3.2
  • Theorem 3.3: characterizations of differentiable convex function on an open set
  • Lemma 3.4: characterization of convex functions via Moreau envelopes
  • Theorem 3.5: characterization of nonsmooth convex functions via subgradient graphical derivatives
  • Example 3.6
  • Corollary 3.7: strong convexity via subgradient graphical derivatives
  • Theorem 4.1: convexity of functions via second subderivatives
  • Corollary 4.2: strong convexity via second subderivatives
  • ...and 3 more