Error bounds for perspective cones of a class of nonnegative Legendre functions
Xiaozhou Wang, Bruno F. Lourenço, Ting Kei Pong
TL;DR
This work analyzes error bounds for conic feasibility problems where the cone is the epigraph of the perspective of a nonnegative Legendre function, with a focus on understanding why the Hölder exponent $\tfrac{1}{2}$ frequently appears. Using facial reduction and the construction of one-step facial residual functions (1-FRFs), the authors show that the 1-FRF for ${\rm epi}\, f^{\pi}$ is Hölderian with exponent $\tfrac{1}{2}$ for almost all boundary points of the dual cone (in 2D Hausdorff measure), which in turn implies that a uniform Hölderian error bound with exponent $\tfrac{1}{2}$ holds generically for a class of feasibility problems involving these cones. The analysis includes a complete facial-structure description of 3D epigraphical cones, explicit 1-FRFs for several exposed-face families, and a measure-theoretic result that excludes a negligibly small set of missing faces, thereby ensuring almost-everywhere applicability. The findings connect to broader implications for convergence rates in conic optimization and related KL-exponent analyses, and they raise open questions about error-bound regularity indices beyond the 1/2 regime, including construction of cones with intermediate EB-rates and extensions to product-cone settings.
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.