Amenable cones: error bounds without constraint qualifications
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.
Keywords
Cite
@article{arxiv.1712.06221,
title = {Amenable cones: error bounds without constraint qualifications},
author = {Bruno F. Lourenço},
journal= {arXiv preprint arXiv:1712.06221},
year = {2021}
}
Comments
36 pages, 1 figure. This version was significantly revised. A discussion on the relation between amenability and related concepts was added. In particular, there is a proof that amenable cones are nice and, therefore, facially exposed. Also, gathered the results on symmetric cones in a single section. Several typos and minor issues were fixed