Lipschitz stability of least-squares problems regularized by functions with $\mathcal{C}^2$-cone reducible conjugates
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910613819621376 |
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| author | Cui, Ying Hoheisel, Tim Nghia, Tran T. A. Sun, Defeng |
| author_facet | Cui, Ying Hoheisel, Tim Nghia, Tran T. A. Sun, Defeng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13118 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lipschitz stability of least-squares problems regularized by functions with $\mathcal{C}^2$-cone reducible conjugates Cui, Ying Hoheisel, Tim Nghia, Tran T. A. Sun, Defeng Optimization and Control 49J52, 49J53, 49K40, 90C25, 90C31 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. |
| title | Lipschitz stability of least-squares problems regularized by functions with $\mathcal{C}^2$-cone reducible conjugates |
| topic | Optimization and Control 49J52, 49J53, 49K40, 90C25, 90C31 |
| url | https://arxiv.org/abs/2409.13118 |