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arXiv · 2409.13118

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

Ying Cui·Tim Hoheisel·Tran T. A. Nghia·Defeng Sun
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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 $\mathcal{C}^2$-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.

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Ying Cui, Tim Hoheisel, Tran T. A. Nghia, Defeng Sun. 2024-09-19. Lipschitz stability of least-squares problems regularized by functions with $\mathcal{C}^2$-cone reducible conjugates. https://arxiv.org/abs/2409.13118

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