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

TL;DR

This paper characterizes the Aubin property of the solution mapping in nonlinear semidefinite programming, establishing necessary and sufficient conditions and connecting it with strong regularity and second-order conditions.

Contribution

It proves the strong second-order sufficient condition is necessary for the Aubin property and links it with strong regularity in NLSDP, advancing understanding of solution stability.

Findings

Strong second-order sufficient condition is necessary for Aubin property.

Several equivalent conditions including strong regularity are established.

Progress in characterizing Aubin property for non-polyhedral constrained problems.

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.