Does CRCQ imply MSCQ in nonpolyhedral conic programming?

Determine whether, for nonpolyhedral conic optimization problems such as second-order cone programs and semidefinite programs, the constant rank constraint qualification (CRCQ) implies the metric subregularity constraint qualification (MSCQ) at a feasible point.

Background

The paper discusses a recent extension of the constant rank constraint qualification (CRCQ) to nonpolyhedral conic programs, including second-order cone and semidefinite programs. While CRCQ is shown to ensure strong second-order necessary conditions and to be independent of Robinson’s constraint qualification, its relationship to the metric subregularity constraint qualification (MSCQ) in the nonpolyhedral setting has not been established.

The authors prove that, for affine second-order cone constraints, CRCQ and MSCQ are equivalent. However, they explicitly note that the broader question—whether CRCQ implies MSCQ in the general nonpolyhedral conic setting—remains unresolved.