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A “joint + marginal” heuristic for 0/1 programs

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  • Jean Lasserre

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  • Tung Thanh

    ()

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    Abstract

    We propose a heuristic for 0/1 programs based on the recent “joint + marginal” approach of the first author for parametric polynomial optimization. The idea is to first consider the n-variable (x 1 , . . . , x n ) problem as a (n − 1)-variable problem (x 2 , . . . , x n ) with the variable x 1 being now a parameter taking value in {0, 1}. One then solves a hierarchy of what we call “joint + marginal” semidefinite relaxations whose duals provide a sequence of polynomial approximations $${x_1\mapsto J_k(x_1)}$$ that converges to the optimal value function J (x 1 ) (as a function of the parameter x 1 ). One considers a fixed index k in the hierarchy and if J k (1) > J k (0) then one decides x 1 = 1 and x 1 =0 otherwise. The quality of the approximation depends on how large k can be chosen (in general, for significant size problems, k=1 is the only choice). One iterates the procedure with now a (n − 2)-variable problem with one parameter $${x_2 \in \{0, 1\}}$$ , etc. Variants are also briefly described as well as some preliminary numerical experiments on the MAXCUT, k-cluster and 0/1 knapsack problems. Copyright Springer Science+Business Media, LLC. 2012

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    File URL: http://hdl.handle.net/10.1007/s10898-011-9788-9
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    Bibliographic Info

    Article provided by Springer in its journal Journal of Global Optimization.

    Volume (Year): 54 (2012)
    Issue (Month): 4 (December)
    Pages: 729-744

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    Handle: RePEc:spr:jglopt:v:54:y:2012:i:4:p:729-744

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    Web page: http://www.springer.com/business/operations+research/journal/10898

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    Related research

    Keywords: 0/1 Programs; Semidefinite relaxations;

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    1. Laurent, M., 2003. "A comparison of the Sherali-Adams, Lov´asz-Schrijver and Lasserre relaxations for 0-1 programming," Open Access publications from Tilburg University urn:nbn:nl:ui:12-3959833, Tilburg University.
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