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 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 provided error bounds for the doubly nonnegative cone.

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

References in corpus (5)

• Sieve-SDP: a simple facial reduction algorithm to preprocess semidefinite programs
• Facial Reduction and Partial Polyhedrality
• The $p$-cones in dimension $n\ge 3$ are not homogeneous when $p\ne 2$
• Optimality conditions for problems over symmetric cones and a simple augmented Lagrangian method
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Cited by in corpus (8)

• Error bounds, facial residual functions and applications to the exponential cone
• Amenable cones are particularly nice
• Convergence analysis under consistent error bounds
• On Positive Duality Gaps in Semidefinite Programming
• Hyperbolicity cones are amenable
• Self-dual polyhedral cones and their slack matrices
• Generalized power cones: optimal error bounds and automorphisms
• A Strict Complementarity Approach to Error Bound and Sensitivity of Solution of Conic Programs