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Second-Order Subdifferential Optimality Conditions in Nonsmooth Optimization

Pham Duy Khanh, Vu Vinh Huy Khoa, Boris S. Mordukhovich, Vo Thanh Phat

TL;DR

The paper advances nonsmooth optimization by deriving new second-order necessary and sufficient optimality conditions expressed through second-order subdifferentials for prox-regular extended-real-valued functions in finite dimensions. It leverages Moreau envelopes to connect local minimizers to smooth surrogate problems, develops a coderivative-based theory of variational (strong) convexity, and establishes neighborhood sufficiency results. The results extend to constrained problems, providing explicit second-order criteria in terms of the problem data under mild constraint qualifications such as MSCQ, and yield novel conditions for NLP, including cases where classical LICQ fails. This framework has potential to improve both theoretical understanding and the design of second-order numerical methods for nonsmooth and constrained optimization, including tilt-stable and strongly minimal solutions.

Abstract

The paper is devoted to deriving novel second-order necessary and sufficient optimality conditions for local minimizers in rather general classes of nonsmooth unconstrained and constrained optimization problems in finite-dimensional spaces. The established conditions are expressed in terms of second-order subdifferentials of lower semicontinuous functions and mainly concern prox-regular objectives that cover a large territory in nonsmooth optimization and its applications. Our tools are based on the machinery of variational analysis and second-order generalized differentiation. The obtained general results are applied to problems of nonlinear programming, where the derived second-order optimality conditions are new even for problems with twice continuously differential data, being expressed there in terms of the classical Hessian matrices.

Second-Order Subdifferential Optimality Conditions in Nonsmooth Optimization

TL;DR

The paper advances nonsmooth optimization by deriving new second-order necessary and sufficient optimality conditions expressed through second-order subdifferentials for prox-regular extended-real-valued functions in finite dimensions. It leverages Moreau envelopes to connect local minimizers to smooth surrogate problems, develops a coderivative-based theory of variational (strong) convexity, and establishes neighborhood sufficiency results. The results extend to constrained problems, providing explicit second-order criteria in terms of the problem data under mild constraint qualifications such as MSCQ, and yield novel conditions for NLP, including cases where classical LICQ fails. This framework has potential to improve both theoretical understanding and the design of second-order numerical methods for nonsmooth and constrained optimization, including tilt-stable and strongly minimal solutions.

Abstract

The paper is devoted to deriving novel second-order necessary and sufficient optimality conditions for local minimizers in rather general classes of nonsmooth unconstrained and constrained optimization problems in finite-dimensional spaces. The established conditions are expressed in terms of second-order subdifferentials of lower semicontinuous functions and mainly concern prox-regular objectives that cover a large territory in nonsmooth optimization and its applications. Our tools are based on the machinery of variational analysis and second-order generalized differentiation. The obtained general results are applied to problems of nonlinear programming, where the derived second-order optimality conditions are new even for problems with twice continuously differential data, being expressed there in terms of the classical Hessian matrices.
Paper Structure (10 sections, 21 theorems, 175 equations, 2 figures)

This paper contains 10 sections, 21 theorems, 175 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries from Variational Analysis
  3. Local Minimizers via Moreau Envelopes
  4. Second-Order Necessary Conditions under Prox-Regularity
  5. Variational Convexity and Second-Order Sufficient Conditions
  6. Second-Order Conditions for Strong Local Minimizers
  7. Second-Order Necessary Conditions in Constrained Optimization
  8. Second-Order Sufficient Conditions in Constrained Optimization
  9. Applications to Nonlinear Programming
  10. Concluding Remarks

Key Result

Proposition 3.1

Let $\varphi\colon{\rm I\!R}^n\to\overline{{\rm I\!R}}$ be an l.s.c. and prox-bounded function that is prox-regular at $\bar{x}$ for $\bar{v}\in\partial\varphi(\bar{x})$ with respect to a radius $\varepsilon>0$. Then for all $\lambda>0$ sufficiently small, we find a convex neighborhood $U_\lambda$ o If $\varphi$ is subdifferentially continuous at $\bar{x}$ for $\bar{v}$, then $T$ in GradEnvelope c

Figures (2)

  • Figure 1: Relationships between second-order optimality conditions, tilt stability, strong variational convexity, and strong local minimizer for continuously prox-regular functions
  • Figure 2: Relationships between constraint qualification conditions

Theorems & Definitions (36)

  • Definition 2.1
  • Definition 2.2
  • Proposition 3.1
  • Theorem 3.2
  • Definition 4.1
  • Remark 4.2
  • Theorem 4.3
  • Remark 4.4
  • Definition 5.1
  • Proposition 5.2
  • ...and 26 more