Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

New second-order optimality conditions for directional optimality of a general set-constrained optimization problem

Wei Ouyang, Jane Ye, Binbin Zhang

TL;DR

The paper addresses second-order optimality conditions for general set-constrained problems with potentially nonconvex feasible sets, focusing on directional local optimality. It develops a framework based on directional MSCQ, directional normal cones, outer second-order tangent sets, and their asymptotic counterparts, together with the lower generalized support function. The authors obtain no-gap necessary and sufficient directional second-order conditions that do not rely on convexity, outer regularity, or nonemptiness of the second-order tangent set, thereby extending prior results. By linking M-multipliers, Clarke multipliers, and directional tangent geometry, the work provides a robust directional-optimality toolkit with potential implications for algorithm design and numerical methods in nonconvex set-constrained optimization.

Abstract

In this paper we derive new second-order optimality conditions for a very general set-constrained optimization problem where the underlying set may be nononvex. We consider local optimality in specific directions (i.e., optimal in a directional neighborhood) in pursuit of developing these new optimality conditions. First-order necessary conditions for local optimality in given directions are provided by virtue of the corresponding directional normal cones. Utilizing the classical and/or the lower generalized support function, we obtain new second-order necessary and sufficient conditions for local optimality of general nonconvex constrained optimization problem in given directions via both the corresponding asymptotic second-order tangent cone and outer second-order tangent set. Our results do not require convexity and/or nonemptyness of the outer second-order tangent set. This is an important improvement to other results in the literature since the outer second-order tangent set can be nonconvex and empty even when the set is convex.

New second-order optimality conditions for directional optimality of a general set-constrained optimization problem

TL;DR

The paper addresses second-order optimality conditions for general set-constrained problems with potentially nonconvex feasible sets, focusing on directional local optimality. It develops a framework based on directional MSCQ, directional normal cones, outer second-order tangent sets, and their asymptotic counterparts, together with the lower generalized support function. The authors obtain no-gap necessary and sufficient directional second-order conditions that do not rely on convexity, outer regularity, or nonemptiness of the second-order tangent set, thereby extending prior results. By linking M-multipliers, Clarke multipliers, and directional tangent geometry, the work provides a robust directional-optimality toolkit with potential implications for algorithm design and numerical methods in nonconvex set-constrained optimization.

Abstract

In this paper we derive new second-order optimality conditions for a very general set-constrained optimization problem where the underlying set may be nononvex. We consider local optimality in specific directions (i.e., optimal in a directional neighborhood) in pursuit of developing these new optimality conditions. First-order necessary conditions for local optimality in given directions are provided by virtue of the corresponding directional normal cones. Utilizing the classical and/or the lower generalized support function, we obtain new second-order necessary and sufficient conditions for local optimality of general nonconvex constrained optimization problem in given directions via both the corresponding asymptotic second-order tangent cone and outer second-order tangent set. Our results do not require convexity and/or nonemptyness of the outer second-order tangent set. This is an important improvement to other results in the literature since the outer second-order tangent set can be nonconvex and empty even when the set is convex.
Paper Structure (7 sections, 19 theorems, 109 equations)

This paper contains 7 sections, 19 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminary Results
  3. Tangent and Normal Cones
  4. Second-Order Tangent Set and Asymptotic Second-Order Tangent Cone to Feasible Set under Directional MSCQ
  5. Optimality Conditions for Nonconvex Optimization Problem with an Abstract Set Constraint
  6. Optimality Conditions for Problem (\ref{['op']})
  7. Conclusion

Key Result

Proposition 2.2

Let $S \subseteq \mathbb{R}^n$ be a closed set, $\bar{x}\in S$ and $d\in T_S(\bar{x})$. Then,

Theorems & Definitions (29)

  • Definition 2.1: Second-Order Tangent Sets shappen98
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Example 2.5
  • Lemma 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Definition 3.1: Directional Local Optimality
  • Proposition 3.2: First-Order Necessary Optimality Condition
  • ...and 19 more