Lipschitz stability of least-squares problems regularized by functions with $\mathcal{C}^2$-cone reducible conjugates
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 -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.
Keywords
Cite
@article{arxiv.2409.13118,
title = {Lipschitz stability of least-squares problems regularized by functions with $\mathcal{C}^2$-cone reducible conjugates},
author = {Ying Cui and Tim Hoheisel and Tran T. A. Nghia and Defeng Sun},
journal= {arXiv preprint arXiv:2409.13118},
year = {2024}
}