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We have decided to discontinue the publication of preprints on our preprint server end of 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
Spectrahedral Containment and Operator Systems with finite-dimensional Realization
Tobias Fritz, Tim Netzer and Andreas ThomAbstract
Containment problems for polytopes and spectrahedra appear in various applications, such as linear and semidefinite programming, combinatorics, convexity and stability analysis of differential equations. This paper explores the theoretical background of a method proposed by Ben-Tal and Nemirovksi [bental]. Their method provides a strengthening of the containment problem, that is algorithmically well tractable.
To analyze this method, we study abstract operator systems, and investigate when they have a finite-dimensional concrete realization.
Our results give some profound insight into their approach. They imply that when testing the inclusion of a fixed polyhedral cone in an arbitrary spectrahedron, the strengthening is tight if and only if the polyhedral cone is a simplex. This is true independent of the representation of the polytope. We also deduce error bounds in the other cases, simplifying and extending recent results by various authors.
- Received:
- 27.09.16
- Published:
- 11.10.16