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Lipschitz Stability of Least-Squares Problems Regularized by Functions with C2-Cone Reducible Conjugates

Ying Cui et al.

Mathematics of Operations Research2026https://doi.org/10.1287/moor.2024.0692article
AJG 3ABDC A
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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 [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].

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https://doi.org/https://doi.org/10.1287/moor.2024.0692

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@article{ying2026,
  title        = {{Lipschitz Stability of Least-Squares Problems Regularized by Functions with C2-Cone Reducible Conjugates}},
  author       = {Ying Cui et al.},
  journal      = {Mathematics of Operations Research},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1287/moor.2024.0692},
}

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