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

TL;DR

This paper establishes the equivalence between the Aubin property and strong regularity in nonlinear second-order cone programming, using a reduction approach and alternative cone choices, advancing variational analysis theory.

Contribution

It introduces a novel reduction approach and cone-based lemma to prove the equivalence, extending previous results to broader SOCP cases.

Findings

Proves the equivalence between Aubin property and strong regularity in nonlinear SOCP.

Provides a new approach to the classical equivalence in nonlinear programming.

Replaces the S-lemma with a cone-based lemma for broader applicability.

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 Ram\'irez [SIAM J. Optim. 21 (2011) 789-823] and Opazo, Outrata, and Ram\'irez [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.