Optimal Error Bounds in the Absence of Constraint Qualifications with Applications to 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 three dimensions that they do not in four or more. Using our error bounds, we analyse least squares problems with p-norm regularization, where our results enable us to compute the corresponding Kurdyka–Łojasiewicz 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. Funding: The second author was partly supported by the Japan Society for the Promotion of Science Grant-in-Aid for Early-Career Scientists [Grants 19K20217, 23K16844] and Grant-in-Aid for Scientific Research [Grant (B)21H03398]. The third author was partly supported by the Hong Kong Research Grants Council [Grant PolyU153000/20p].