A Complete Characterization of the Robust Isolated Calmness of Nuclear Norm Regularized Convex Optimization Problems (Journal of Computational Mathematics, 2018)
In this paper, we provide a complete characterization of the robust isolated calmnessof the Karush-Kuhn-Tucker (KKT) solution mapping for convex constrained optimizationproblems regularized by the nuclear norm function. This study is motivated by the recentwork in [8], where the authors show that under the Robinson constraint qualification at alocal optimal solution, the KKT solution mapping for a wide class of conic programmingproblems is robustly isolated calm if and only if both the second order sufficient condition(SOSC) and the strict Robinson constraint qualification (SRCQ) are satisfied. Based onthe variational properties of the nuclear norm function and its conjugate, we establishthe equivalence between the primal/dual SOSC and the dual/primal SRCQ. The derivedresults lead to several equivalent characterizations of the robust isolated calmness of theKKT solution mapping and add insights to the existing literature on the stability of nuclearnorm regularized convex optimization problems.
Keywords: Robust isolated calmness; Nuclear norm; Second order sufficient condition; Strict Robinson constraint qualification.