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Unifying semidefinite and set-copositive relaxations of binary problems and randomization techniques

Author

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  • Felix Lieder
  • Fatemeh Rad
  • Florian Jarre

Abstract

A reformulation of quadratically constrained binary programs as duals of set-copositive linear optimization problems is derived using either $$\{0,1\}$$ { 0 , 1 } -formulations or $$\{-1,1\}$$ { - 1 , 1 } -formulations. The latter representation allows an extension of the randomization technique by Goemans and Williamson. An application to the max-clique problem shows that the max-clique problem is equivalent to a linear program over the max-cut polytope with one additional linear constraint. This transformation allows the solution of a semidefinite relaxation of the max-clique problem with about the same computational effort as the semidefinite relaxation of the max-cut problem—independent of the number of edges in the underlying graph. A numerical comparison of this approach to the standard Lovasz number concludes the paper. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Felix Lieder & Fatemeh Rad & Florian Jarre, 2015. "Unifying semidefinite and set-copositive relaxations of binary problems and randomization techniques," Computational Optimization and Applications, Springer, vol. 61(3), pages 669-688, July.
    • Handle: RePEc:spr:coopap:v:61:y:2015:i:3:p:669-688
    DOI: 10.1007/s10589-015-9731-y
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    Cited by:

    1. Florian Jarre & Felix Lieder & Ya-Feng Liu & Cheng Lu, 2020. "Set-completely-positive representations and cuts for the max-cut polytope and the unit modulus lifting," Journal of Global Optimization, Springer, vol. 76(4), pages 913-932, April.
    2. Jose L. Walteros & Austin Buchanan, 2020. "Why Is Maximum Clique Often Easy in Practice?," Operations Research, INFORMS, vol. 68(6), pages 1866-1895, November.

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