Lipschitz stability of least-squares problems regularized by functions with $\mathcal{C}^2$-cone reducible conjugates

TL;DR

This paper establishes new conditions for Lipschitz stability of solutions in regularized least-squares problems with convex regularizers having $\ ext{C}^2$-cone reducible conjugates, using dual regularity and first-order analysis.

Contribution

It introduces a novel approach based on Robinson's strong regularity to characterize Lipschitz stability without second-order curvature information.

Findings

Solution mappings are Lipschitz continuous when locally single-valued.

New characterizations of full stability and tilt stability are provided.

Approach applies to a broad class of convex additive-composite problems.

Abstract

In this paper, we study Lipschitz continuity of the solution mappings of regularized least-squares problems for which the convex regularizers have (Fenchel) conjugates that are $\mathcal{C}^2$-cone reducible. Our approach, by using Robinson's strong regularity on the dual problem, allows us to obtain new characterizations of Lipschitz stability that rely solely on first-order information, thus bypassing the need to explore second-order information (curvature) of the regularizer. We show that these solution mappings are automatically Lipschitz continuous around the points in question whenever they are locally single-valued. We leverage our findings to obtain new characterizations of full stability and tilt stability for a broader class of convex additive-composite problems.