The robust isolated calmness of spectral norm regularized convex matrix optimization problems

Abstract This article aims to provide a series of characterizations of the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) mapping for spectral norm regularized convex optimization problems. By establishing the variational properties of the spectral norm function, we directly prove that the KKT mapping is isolated calm if and only if the strict Robinson constraint qualification (SRCQ) and the second-order sufficient condition (SOSC) hold. Furthermore, we obtain the crucial result that the SRCQ for the primal/dual problem and the SOSC for the dual/primal problem are equivalent. The obtained results can derive more equivalent conditions of the robust isolated calmness of the KKT mapping, thereby enriching the stability theory of spectral norm regularized optimization problems and enhancing the usability of isolated calmness in algorithm applications.