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Characterizations of the Aubin Property of the Solution Mapping for Nonlinear Semidefinite Programming

Liang Chen, Ruoning Chen, Defeng Sun, Liping Zhang

TL;DR

This work addresses the Aubin property of the KKT solution mapping for nonlinear semidefinite programming (NLSDP) at a locally optimal point. By employing the Mordukhovich criterion and a novel reduction strategy, it shows that the Aubin property is equivalent to the strong second-order sufficient condition (SSOSC) together with constraint nondegeneracy, and it establishes a network of equivalent characterizations including strong regularity and strong metric regularity for NLSDP. Building on explicit coderivative formulas for the positive semidefinite cone, the paper extends existing results from NLP and NLSOCP to NLSDP and relates these findings to general non-polyhedral $C^2$-cone reducible problems. The methodology clarifies when the KKT system behaves in a Lipschitz-stable manner under perturbations, with implications for sensitivity analysis and algorithmic design in NLSDP contexts.

Abstract

In this paper, we study the Aubin property of the Karush-Kuhn-Tucker solution mapping for the nonlinear semidefinite programming (NLSDP) problem at a locally optimal solution. In the literature, it is known that the Aubin property implies the constraint nondegeneracy by Fusek [SIAM J. Optim. 23 (2013), pp. 1041-1061] and the second-order sufficient condition by Ding et al. [SIAM J. Optim. 27 (2017), pp. 67-90]. Based on the Mordukhovich criterion, here we further prove that the strong second-order sufficient condition is also necessary for the Aubin property to hold. Consequently, several equivalent conditions including the strong regularity are established for NLSDP's Aubin property. Together with the recent progress made by Chen et al. on the equivalence between the Aubin property and the strong regularity for nonlinear second-order cone programming [SIAM J. Optim., in press; arXiv:2406.13798v3 (2024)], this paper constitutes a significant step forward in characterizing the Aubin property for general non-polyhedral $C^2$-cone reducible constrained optimization problems.

Characterizations of the Aubin Property of the Solution Mapping for Nonlinear Semidefinite Programming

TL;DR

This work addresses the Aubin property of the KKT solution mapping for nonlinear semidefinite programming (NLSDP) at a locally optimal point. By employing the Mordukhovich criterion and a novel reduction strategy, it shows that the Aubin property is equivalent to the strong second-order sufficient condition (SSOSC) together with constraint nondegeneracy, and it establishes a network of equivalent characterizations including strong regularity and strong metric regularity for NLSDP. Building on explicit coderivative formulas for the positive semidefinite cone, the paper extends existing results from NLP and NLSOCP to NLSDP and relates these findings to general non-polyhedral -cone reducible problems. The methodology clarifies when the KKT system behaves in a Lipschitz-stable manner under perturbations, with implications for sensitivity analysis and algorithmic design in NLSDP contexts.

Abstract

In this paper, we study the Aubin property of the Karush-Kuhn-Tucker solution mapping for the nonlinear semidefinite programming (NLSDP) problem at a locally optimal solution. In the literature, it is known that the Aubin property implies the constraint nondegeneracy by Fusek [SIAM J. Optim. 23 (2013), pp. 1041-1061] and the second-order sufficient condition by Ding et al. [SIAM J. Optim. 27 (2017), pp. 67-90]. Based on the Mordukhovich criterion, here we further prove that the strong second-order sufficient condition is also necessary for the Aubin property to hold. Consequently, several equivalent conditions including the strong regularity are established for NLSDP's Aubin property. Together with the recent progress made by Chen et al. on the equivalence between the Aubin property and the strong regularity for nonlinear second-order cone programming [SIAM J. Optim., in press; arXiv:2406.13798v3 (2024)], this paper constitutes a significant step forward in characterizing the Aubin property for general non-polyhedral -cone reducible constrained optimization problems.
Paper Structure (10 sections, 13 theorems, 125 equations)

This paper contains 10 sections, 13 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Coderivative related to positive semidefinite cone
  4. Second-order sufficient conditions
  5. Technical lemmas
  6. Implications of the Aubin property for NLSDP
  7. A reduction method for NLSDP
  8. Aubin property implies SSOSC
  9. Characterizations of the Aubin property for NLSDP
  10. Conclusions

Key Result

Lemma 2.1

sdpcod Suppose that $A\in {\mathcal{S}}^{p}$ has the eigenvalue decomposition in eigd. Then, $U\in{\mathcal{D}}^*{\mathcal{N}}_{{\mathcal{S}}_+^p}(A_+, A_-)(V)$ if and only if with where $\widetilde{U}:=P^\top UP\in{\mathcal{S}}^p$, $\widetilde{V}:=P^\top VP\in{\mathcal{S}}^p$, and $\Sigma\in{\mathds R}^{p\times p}$ is the matrix defined by where $0/0$ is defined to be $1$. In addition, $\wide

Theorems & Definitions (32)

  • Lemma 2.1
  • Remark 2.1
  • Remark 2.2
  • Example 2.1
  • Definition 2.1
  • Remark 2.3
  • Lemma 2.2
  • Lemma 2.3: varbook
  • Lemma 2.4: magz
  • Lemma 2.5
  • ...and 22 more