A complete characterization on the robust isolated calmness of the nuclear norm regularized convex optimization problems

TL;DR

This paper provides a comprehensive characterization of the conditions under which the KKT solution mapping for nuclear norm regularized convex problems is robustly isolated calm, enhancing understanding of stability in such optimization problems.

Contribution

It establishes the equivalence between primal/dual SOSC and dual/primal SRCQ for nuclear norm regularized problems, offering new insights into their stability properties.

Findings

Equivalent characterizations of robust isolated calmness are derived.

The variational properties of the nuclear norm function are utilized.

The results extend existing stability analysis to a broader class of problems.

Abstract

In this paper, we provide a complete characterization on the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) solution mapping for convex constrained optimization problems regularized by the nuclear norm function. This study is motivated by the recent work in [8], where the authors show that under the Robinson constraint qualification at a local optimal solution, the KKT solution mapping for a wide class of conic programming problem 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 on the variational properties of the nuclear norm function and its conjugate, we establish the equivalence between the primal/dual SOSC and the dual/primal SRCQ. The derived results lead to several equivalent characterizations of the robust isolated calmness of the KKT solution mapping and add insights to the existing literature on the stability of the nuclear norm regularized convex optimization problems.