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Symmetry in Mathematical Programming

Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (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.

Key words

MINLP NLP reformulation group graph isomorphism permutation expression tree 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.LIX, École PolytechniquePalaiseauFrance

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