Aubin Property and Strong Regularity Are Equivalent for Nonlinear Second-Order Cone Programming

TL;DR

The paper resolves the open problem of whether the Aubin property of the perturbed KKT solution map ${{\mathbb S}}_{\rm KKT}$ and the strong regularity of the KKT system are equivalent for nonlinear SOCP at a locally optimal solution, without assuming convexity or strict complementarity. It introduces a reduction based on the Mordukhovich criterion and a cone-alternative lemma to replace the S-lemma, and proves that Aubin property is equivalent to strong regularity for the corresponding generalized equation, ultimately deriving the strong second-order sufficient condition. The authors provide a detailed coderivative analysis of second-order cones, construct a dimension-reduction framework via a sequence of matrices ($\Theta$, $H$, $R_i$, $T_i$), and show that the resulting SSOSC holds under the Aubin property, establishing a robust equivalence with broad implications. As a byproduct, the work extends the Dontchev–Rockafellar result from conventional nonlinear programming to nonlinear SOCP with $C^2$-cone reducible constraints and discusses consequences such as nondegeneracy, full stability, and nonsingularity of Clarke-type Jacobians. These results advance our understanding of variational properties in nonpolyhedral conic programming and suggest potential extensions to more general conic settings.

Abstract

This paper solves a fundamental open problem in variational analysis on the equivalence between the Aubin property and the strong regularity for nonlinear second-order cone programming (SOCP) at a locally optimal solution. We achieve this by introducing a reduction approach to the Aubin property characterized by the Mordukhovich criterion and a lemma of alternative choices on cones to replace the S-lemma used in Outrata and Ramírez [SIAM J. Optim. 21 (2011) 789-823] and Opazo, Outrata, and Ramírez [SIAM J. Optim. 27 (2017) 2141-2151], where the same SOCP was considered under the strict complementarity condition except for possibly only one block of constraints. As a byproduct, we also offer a new approach to the well-known result of Dontchev and Rockafellar [SIAM J. Optim. 6 (1996) 1087-1105] on the equivalence of the two concepts in conventional nonlinear programming.

Theorems & Definitions (29)

• Lemma 2.1
• Proof 1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 3.1
• Lemma 3.2
• Proof 2
• Definition 4.1