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Global quadratic optimization via conic relaxation

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  • NESTEROV, Yurii

    (Center for Operations Research and Econometrics (CORE), Université catholique de Louvain (UCL), Louvain la Neuve, Belgium)

Abstract

We present a convex conic relaxation for a problem of maximizing an indefinite quadratic form over a set of convex constraints on the squared variables. We show that for all these problems we get at least 12/37-relative accuracy of the approximation. In the second part of the paper we derive the conic relaxation by another approach based on the second order optimality conditions. We show that for lp-balls, p>=2, intersected by a linear subspace, it is possible to guarantee (1- 2/p)-relative accuracy of the solution. As a consequence, we prove (1 - 1/eln-n)-relative accuracy of the conic relaxation for an indefinite quadratic maximization problem over an n-dimensional unit box with homogeneous linear equality constraints. We discuss the implications of the results for the discussion around the question P = NP.

Suggested Citation

  • NESTEROV, Yurii, 1998. "Global quadratic optimization via conic relaxation," LIDAM Discussion Papers CORE 1998060, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:1998060
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    File URL: https://sites.uclouvain.be/core/publications/coredp/coredp1998.html
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 1997. "Semidefinite relaxation and nonconvex quadratic optimization," LIDAM Discussion Papers CORE 1997044, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. NESTEROV, Yurii, 1997. "Quality of semidefinite relaxation for nonconvex quadratic optimization," LIDAM Discussion Papers CORE 1997019, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Alexandre d'Aspremont & Noureddine El Karoui, 2013. "Weak Recovery Conditions from Graph Partitioning Bounds and Order Statistics," Mathematics of Operations Research, INFORMS, vol. 38(2), pages 228-247, May.
    2. NESTEROV, Yurii, 1999. "Global quadratic optimization on the sets with simplex structure," LIDAM Discussion Papers CORE 1999015, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Guy Kindler & Assaf Naor & Gideon Schechtman, 2010. "The UGC Hardness Threshold of the L p Grothendieck Problem," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 267-283, May.

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