{"ID":2862809,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.26289","arxiv_id":"2509.26289","title":"Error bounds for perspective cones of a class of nonnegative Legendre functions","abstract":"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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2509.26289","url_pdf":"https://arxiv.org/pdf/2509.26289v1","authors":"[\"Xiaozhou Wang\",\"Bruno F. Lourenço\",\"Ting Kei Pong\"]","published":"2025-09-30T14:07:37Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.MG\",\"math.NA\"]","methods":"[]","has_code":false}