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

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.