Optimal error bounds in the absence of constraint qualifications with applications to the $p$-cones and beyond
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866913287020478464 |
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| author | Lindstrom, Scott B. Lourenço, Bruno F. Pong, Ting Kei |
| author_facet | Lindstrom, Scott B. Lourenço, Bruno F. Pong, Ting Kei |
| contents | We prove tight Hölderian error bounds for all $p$-cones. Surprisingly, the exponents differ in several ways from those that have been previously conjectured; moreover, they illuminate $p$-cones as a curious example of a class of objects that possess properties in 3 dimensions that they do not in 4 or more. Using our error bounds, we analyse least squares problems with $p$-norm regularization, where our results enable us to compute the corresponding KL exponents for previously inaccessible values of $p$. Another application is a (relatively) simple proof that most $p$-cones are neither self-dual nor homogeneous. Our error bounds are obtained under the framework of facial residual functions, and we expand it by establishing for general cones an optimality criterion under which the resulting error bound must be tight. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_11729 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Optimal error bounds in the absence of constraint qualifications with applications to the $p$-cones and beyond Lindstrom, Scott B. Lourenço, Bruno F. Pong, Ting Kei Optimization and Control Numerical Analysis 90C25, 52A20 We prove tight Hölderian error bounds for all $p$-cones. Surprisingly, the exponents differ in several ways from those that have been previously conjectured; moreover, they illuminate $p$-cones as a curious example of a class of objects that possess properties in 3 dimensions that they do not in 4 or more. Using our error bounds, we analyse least squares problems with $p$-norm regularization, where our results enable us to compute the corresponding KL exponents for previously inaccessible values of $p$. Another application is a (relatively) simple proof that most $p$-cones are neither self-dual nor homogeneous. Our error bounds are obtained under the framework of facial residual functions, and we expand it by establishing for general cones an optimality criterion under which the resulting error bound must be tight. |
| title | Optimal error bounds in the absence of constraint qualifications with applications to the $p$-cones and beyond |
| topic | Optimization and Control Numerical Analysis 90C25, 52A20 |
| url | https://arxiv.org/abs/2109.11729 |