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Semidefinite relaxations for partitioning, assignment and ordering problems

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Abstract

Semidefinite optimization is a strong tool in the study of NP-hard combinatorial optimization problems. On the one hand, semidefinite optimization problems are in principle solvable in polynomial time (with fixed precision), on the other hand, their modeling power allows to naturally handle quadratic constraints. Contrary to linear optimization with the efficiency of the Simplex method, the algorithmic treatment of semidefinite problems is much more subtle and also practically quite expensive. This survey-type article is meant as an introduction for a non-expert to this exciting area. The basic concepts are explained on a mostly intuitive level, and pointers to advanced topics are given. We provide a variety of semidefinite optimization models on a selection of graph optimization problems and give a flavour of their practical impact.

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Notes

  1. http://plato.asu.edu/bench.

  2. http://www.seas.upenn.edu/qaplib/.

  3. http://dimacs.rutgers.edu/Challenges/.

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Acknowledgments

I thank Miguel Anjos and an anonymous referee for giving numerous suggestions to improve the presentation.

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Authors and Affiliations

  1. Institut für Mathematik, Alpen-Adria Universität Klagenfurt, 9020, Klagenfurt, Austria

    F. Rendl

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  1. F. Rendl
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Correspondence to F. Rendl.

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This is an updated version of the paper that appeared in 4OR, 10, 321–346, (2012), see Rendl (2012).

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Rendl, F. Semidefinite relaxations for partitioning, assignment and ordering problems. Ann Oper Res 240, 119–140 (2016). https://doi.org/10.1007/s10479-015-2015-1

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