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Symmetry in Mathematical Programming

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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

Symmetry is mainly exploited in mathematical programming in order to reduce the computation times of enumerative algorithms. The most widespread approach rests on: (a) finding symmetries in the problem instance; (b) reformulating the problem so that it does not allow some of the symmetric optima; (c) solving the modified problem. Sometimes (b) and (c) are performed concurrently: the solution algorithm generates a sequence of subproblems, some of which are recognized to be symmetrically equivalent and either discarded or treated differently. We review symmetry-based analyses and methods for Linear Programming, Integer Linear Programming, Mixed-Integer Linear Programming and Semidefinite Programming. We then discuss a method (introduced in [36]) for automatically detecting symmetries of general (nonconvex) Nonlinear and Mixed-Integer Nonlinear Programming problems and a reformulation based on adjoining symmetry breaking constraints to the original formulation. We finally present a new theoretical and computational study of the formulation symmetries of the Kissing Number Problem.

AMS(MOS) subject classifications. 90C11, 90C26, 90C30, 05C25, 20B25.

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Authors and Affiliations

  1. LIX, École Polytechnique, F-91128, Palaiseau, France

    Leo Liberti

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  1. Leo Liberti
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Correspondence to Leo Liberti .

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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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Liberti, L. (2012). Symmetry in Mathematical Programming. 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_9

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