Amenable cones: error bounds without constraint qualifications

TL;DR

This paper introduces a new framework for deriving error bounds in linear conic problems without the need for constraint qualifications, using the concepts of amenable cones and facial residual functions.

Contribution

It develops the notion of amenable cones and facial residual functions, extending error bounds to cases without regularity assumptions, including symmetric and doubly nonnegative cones.

Findings

Symmetric cones are shown to be amenable.

A new Hölderian error bound is established.

Error bounds are provided for intersections of amenable cones.

Abstract

We provide a framework for obtaining error bounds for linear conic problems without assuming constraint qualifications or regularity conditions. The key aspects of our approach are the notions of amenable cones and facial residual functions. For amenable cones, it is shown that error bounds can be expressed as a composition of facial residual functions. The number of compositions is related to the facial reduction technique and the singularity degree of the problem. In particular, we show that symmetric cones are amenable and compute facial residual functions. From that, we are able to furnish a new H\"olderian error bound, thus extending and shedding new light on an earlier result by Sturm on semidefinite matrices. We also provide error bounds for the intersection of amenable cones, this will be used to provided error bounds for the doubly nonnegative cone.