Lipschitz Stability of Least-Squares Problems Regularized by Functions with ${\mathcal{C}}^2$-Cone Reducible Conjugates

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.

Funding: Y. Cui is partially supported by the National Science Foundation [Grant DMS-2416250] and the National Institutes of Health [Grant 1R01CA287413-01]. T. Hoheisel is supported by an NSERC Discovery grant [Grant RGPIN-2024-04116]. The research of D. Sun was supported in part by the Hong Kong Research Grants Council [Grant GRF project 15309625] and the RGC Senior Research Fellow scheme [Grant SRFS2223-5S02].