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 Formula: see text-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.
Cui et al. (Mon,) studied this question.
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