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Strong duality and minimal representations for cone optimization

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  • Levent Tunçel
  • Henry Wolkowicz

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    Abstract

    The elegant theoretical results for strong duality and strict complementarity for linear programming, LP, lie behind the success of current algorithms. In addition, preprocessing is an essential step for efficiency in both simplex type and interior-point methods. However, the theory and preprocessing techniques can fail for cone programming over nonpolyhedral cones. We take a fresh look at known and new results for duality, optimality, constraint qualifications, CQ, and strict complementarity, for linear cone optimization problems in finite dimensions. One theme is the notion of minimal representation of the cone and the constraints. This provides a framework for preprocessing cone optimization problems in order to avoid both the theoretical and numerical difficulties that arise due to the (near) loss of the strong CQ, strict feasibility. We include results and examples on the surprising theoretical connection between duality gaps in the original primal-dual pair and lack of strict complementarity in their homogeneous counterpart. Our emphasis is on results that deal with Semidefinite Programming, SDP. Copyright Springer Science+Business Media, LLC 2012

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    File URL: http://hdl.handle.net/10.1007/s10589-012-9480-0
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    Bibliographic Info

    Article provided by Springer in its journal Computational Optimization and Applications.

    Volume (Year): 53 (2012)
    Issue (Month): 2 (October)
    Pages: 619-648

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    Handle: RePEc:spr:coopap:v:53:y:2012:i:2:p:619-648

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    Web page: http://www.springer.com/math/journal/10589

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    Related research

    Keywords: Cone optimization; Duality; Preprocessing; Constraint qualification; Duality gap; Semidefinite programming; Strict complementarity; Nice cones; Devious cones; Facially dual complete cones;

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    1. Jibetean, D. & Laurent, M., 2005. "Semidefinite approximations for global unconstrained polynomial optimization," Open Access publications from Tilburg University urn:nbn:nl:ui:12-3959914, Tilburg University.
    2. Halická, M. & Klerk, E. de & Roos, C., 2002. "On the convergence of the central path in semidefinite optimization," Open Access publications from Tilburg University urn:nbn:nl:ui:12-226112, Tilburg University.
    3. Helmberg, C., 2002. "Semidefinite programming," European Journal of Operational Research, Elsevier, vol. 137(3), pages 461-482, March.
    4. Maria Gonzalez-Lima & Hua Wei & Henry Wolkowicz, 2009. "A stable primal–dual approach for linear programming under nondegeneracy assumptions," Computational Optimization and Applications, Springer, vol. 44(2), pages 213-247, November.
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