Optimal error bounds in the absence of constraint qualifications with applications to the $p$-cones and beyond

Abstract

We prove tight Hölderian error bounds for all p-cones. Surprisingly, the exponents differ in several ways from those that have been previously conjectured; moreover, they illuminate p-cones as a curious example of a class of objects that possess properties in 3 dimensions that they do not in 4 or more. Using our error bounds, we analyse least squares problems with pnorm regularization, where our results enable us to compute the corresponding KL exponents for previously inaccessible values of p. Another application is a (relatively) simple proof that most p-cones are neither self-dual nor homogeneous. Our error bounds are obtained under the framework of facial residual functions and we expand it by establishing for general cones an optimality criterion under which the resulting error bound must be tight.

Facial reduction for the Shor SDP relaxation of QCQPs

“…Sturm's error bound is very influential, and many exciting new results are developed subsequently. For example, Sturm's bound is extended and applied to more interesting cases in [17,18,19]. In [20], the bound is used to explain how the exponential size solutions arise in SDP.…”