Revisiting the Constant-Rank Constraint Qualification for Second-Order Cone Programs

TL;DR

This paper investigates the constant rank constraint qualification (CRCQ) for second-order cone programs in the linear case, revealing its limitations, providing a characterization, and establishing its equivalence with the metric subregularity constraint qualification (MSCQ).

Contribution

It offers a necessary and sufficient condition for CRCQ's facial constant rank property and proves CRCQ and MSCQ are equivalent in this context.

Findings

Facial constant rank property does not always hold in linear second-order cone programs.

A new characterization of CRCQ is derived that is easily verifiable.

CRCQ and MSCQ are shown to be equivalent for linear second-order cone programs.

Abstract

The constant rank constraint qualification (CRCQ) for second-order cone programs, introduced by Andreani et al. in [Math. Program. 202 (2023), 473 - 513], shares some desirable properties with its classical nonlinear programming counterpart; specifically, it guarantees strong second-order necessary conditions for optimality, and is independent of the Robinson constraint qualification. However, unlike the classical version, this new CRCQ can fail in the linear case, and it is unclear whether CRCQ implies the metric subregularity constraint qualification (MSCQ). The aim of this paper is to examine the CRCQ for second-order cone programs in the linear setting. First, we show that the facial constant rank property, which is a key requirement for the validity of CRCQ, does not always hold in this context. Then, we derive a necessary and sufficient condition for a feasible point to satisfy this property. After that, we establish an easily verifiable characterization of CRCQ. Finally, utilizing this characterization, we prove that CRCQ and MSCQ are equivalent.