A comprehensive characterization of the robust isolated calmness of Ky Fan $ k $-norm regularized convex matrix optimization problems

Abstract

This paper extends a result of isolated calmness for nuclear norm regularized convex optimization problems to Ky Fan $ k $-norm regularized convex optimization problems. We find that there exists a certain equivalence relationship among the critical cones of the Ky Fan $ k $-norm function and its conjugate as well as the 'sigma term', namely, the conjugate function of the parabolic second-order directional derivative of the Ky Fan $ k $-norm. By establishing the equivalence between the primal (dual) strict Robinson constraint qualification (SRCQ) and the dual (primal) second-order sufficient condition (SOSC), we derive a series of complete characterizations of the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) mapping for Ky Fan $ k $-norm regularized convex matrix optimization problems. The obtained results enrich the stability theory of the Ky Fan $ k $-norm regularized convex optimization problems and further enhance the usability of the related algorithms.

Keywords

• isolated calmness
• ky fan $ k $-norm
• critical cone
• second-order sufficient condition
• strict robinson constraint qualification