Amenable cones: error bounds without constraint qualifications

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...

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Bibliographic Details
Published in:Mathematical programming 2021-03, Vol.186 (1-2), p.1-48
Main Author: Lourenço, Bruno F.
Format: Article
Language:English
Subjects:
Calculus of Variations and Optimal Control
Optimization
Combinatorics
Composition
Cones
Errors
Full Length Paper
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Matrix methods
Numerical Analysis
Theoretical
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Summary: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 Hölderian 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 prove error bounds for the doubly nonnegative cone. At the end, we list some open problems.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-019-01439-3