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Matrix Relaxations in Combinatorial Optimization

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Mixed Integer Nonlinear Programming

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 154))

Abstract

The success of interior-point methods to solve semidefinite optimization problems (SDP) has spurred interest in SDP as a modeling tool in various mathematical fields, in particular in combinatorial optimization. SDP can be viewed as a matrix relaxation of the underlying 0-1 optimization problem. In this survey, some of the main techniques to get matrix relaxations of combinatorial optimization problems are presented. These are based either on semidefinite matrices, leading in general to tractable relaxations, or on completely positive or copositive matrices. Copositive programs are intractable, but they can be used to get exact formulations of many NP-hard combinatorial optimization problems. It is the purpose of this survey to show the potential of matrix relaxations.

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Authors and Affiliations

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

    Franz Rendl

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  1. Franz Rendl
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  1. , College of Engineering, University of Michigan, 2355 Bonisteel Boulevard, Ann Arbor, 48109, Michigan, USA

    Jon Lee

  2. , Mathematics and Computer Science, Argonne National Laboratory, Argonne, 60439, Illinois, USA

    Sven Leyffer

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Rendl, F. (2012). Matrix Relaxations in Combinatorial Optimization. In: Lee, J., Leyffer, S. (eds) Mixed Integer Nonlinear Programming. The IMA Volumes in Mathematics and its Applications, vol 154. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1927-3_17

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