Isolated Calmness in Regularized Convex Optimization

TL;DR

This paper investigates the isolated calmness property in regularized convex optimization, providing geometric conditions and a new zero-product property for second-order structures to facilitate analysis.

Contribution

It introduces necessary and sufficient geometric conditions for isolated calmness and develops a zero-product property for second-order structures in convex optimization.

Findings

Established simple geometric conditions for isolated calmness.

Developed a zero-product property for second-order structures.

Provided criteria that are easy to verify in practice.

Abstract

This paper studies the isolated calmness of the optimal solution mapping and the associated Lagrange system for regularized convex composite optimization problems. Several necessary and sufficient conditions for this property are established. These conditions are geometric in nature and relatively simple to verify. To support the analysis, we also develop a so-called zero-product property for second-order structures, namely the graphical derivative of the subgradient mapping of convex functions.