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

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.