Error bounds for perspective cones of a class of nonnegative Legendre functions

2025/09/30 by Wang, Xiaozhou, Lourenço, Bruno F., Pong, Ting Kei
#FOS: Mathematics #Metric Geometry (math.MG) #Numerical Analysis (math.NA) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2509.26289

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 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.

Citations

• Clarabel: An interior-point solver for conic programs with quadratic objectives
• Tight error bounds for log-determinant cones without constraint qualifications
• Optimal error bounds in the absence of constraint qualifications with applications to the p-cones and beyond
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