Error bounds for perspective cones of a class of nonnegative Legendre functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916980205813760 |
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| author | Wang, Xiaozhou Lourenço, Bruno F. Pong, Ting Kei |
| author_facet | Wang, Xiaozhou Lourenço, Bruno F. Pong, Ting Kei |
| contents | Error bounds play a central role in the study of conic optimization problems, including the analysis of convergence rates for numerous algorithms. Curiously, those error bounds are often Hölderian with exponent 1/2. In this paper, we try to explain the prevalence of the 1/2 exponent by investigating generic properties of error bounds for conic feasibility problems where the underlying cone is a perspective cone constructed from a nonnegative Legendre function on $\mathbb{R}$. Our analysis relies on the facial reduction technique and the computation of one-step facial residual functions (1-FRFs). Specifically, under appropriate assumptions on the Legendre function, we show that 1-FRFs can be taken to be Hölderian of exponent 1/2 almost everywhere with respect to the two-dimensional Hausdorff measure. This enables us to further establish that having a uniform Hölderian error bound with exponent 1/2 is a generic property for a class of feasibility problems involving these cones. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_26289 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Error bounds for perspective cones of a class of nonnegative Legendre functions Wang, Xiaozhou Lourenço, Bruno F. Pong, Ting Kei Optimization and Control Numerical Analysis Metric Geometry Error bounds play a central role in the study of conic optimization problems, including the analysis of convergence rates for numerous algorithms. Curiously, those error bounds are often Hölderian with exponent 1/2. In this paper, we try to explain the prevalence of the 1/2 exponent by investigating generic properties of error bounds for conic feasibility problems where the underlying cone is a perspective cone constructed from a nonnegative Legendre function on $\mathbb{R}$. Our analysis relies on the facial reduction technique and the computation of one-step facial residual functions (1-FRFs). Specifically, under appropriate assumptions on the Legendre function, we show that 1-FRFs can be taken to be Hölderian of exponent 1/2 almost everywhere with respect to the two-dimensional Hausdorff measure. This enables us to further establish that having a uniform Hölderian error bound with exponent 1/2 is a generic property for a class of feasibility problems involving these cones. |
| title | Error bounds for perspective cones of a class of nonnegative Legendre functions |
| topic | Optimization and Control Numerical Analysis Metric Geometry |
| url | https://arxiv.org/abs/2509.26289 |