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Alternative SDP and SOCP approximations for polynomial optimization

Author

Listed:
  • Xiaolong Kuang

    (Lehigh University)

  • Bissan Ghaddar

    (University of Western Ontario)

  • Joe Naoum-Sawaya

    (University of Western Ontario)

  • Luis F. Zuluaga

    (Lehigh University)

Abstract

In theory, hierarchies of semidefinite programming (SDP) relaxations based on sum of squares (SOS) polynomials have been shown to provide arbitrarily close approximations for a general polynomial optimization problem (POP). However, due to the computational challenge of solving SDPs, it becomes difficult to use SDP hierarchies for large-scale problems. To address this, hierarchies of second-order cone programming (SOCP) relaxations resulting from a restriction of the SOS polynomial condition have been recently proposed to approximate POPs. Here, we consider alternative ways to use the SOCP restrictions of the SOS condition. In particular, we show that SOCP hierarchies can be effectively used to strengthen hierarchies of linear programming relaxations for POPs. Specifically, we show that this solution approach is substantially more effective in finding solutions of certain POPs for which the more common hierarchies of SDP relaxations are known to perform poorly. Furthermore, when the feasible set of the POP is compact, these SOCP hierarchies converge to the POP’s optimal value.

Suggested Citation

  • Xiaolong Kuang & Bissan Ghaddar & Joe Naoum-Sawaya & Luis F. Zuluaga, 2019. "Alternative SDP and SOCP approximations for polynomial optimization," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 7(2), pages 153-175, June.
    • Handle: RePEc:spr:eurjco:v:7:y:2019:i:2:d:10.1007_s13675-018-0101-2
    DOI: 10.1007/s13675-018-0101-2
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    References listed on IDEAS

    as
    1. Peter Dickinson & Janez Povh, 2015. "On an extension of Pólya’s Positivstellensatz," Journal of Global Optimization, Springer, vol. 61(4), pages 615-625, April.
    2. de Klerk, E. & Sotirov, R., 2010. "Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem," Other publications TiSEM 73287c80-3bc2-40c4-b02d-4, Tilburg University, School of Economics and Management.
    3. Jean B. Lasserre & Kim-Chuan Toh & Shouguang Yang, 2017. "A bounded degree SOS hierarchy for polynomial optimization," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 5(1), pages 87-117, March.
    4. de Klerk, E. & Sotirov, R., 2007. "Exploiting Group Symmetry in Semidefinite Programming Relaxations of the Quadratic Assignment Problem," Other publications TiSEM 87a5d126-86e5-4863-8ea5-1, Tilburg University, School of Economics and Management.
    5. de Klerk, E. & Sotirov, R., 2007. "Exploiting Group Symmetry in Semidefinite Programming Relaxations of the Quadratic Assignment Problem," Other publications TiSEM 87a5d126-86e5-4863-8ea5-1, Tilburg University, School of Economics and Management.
    6. Jean B. Lasserre, 2002. "Semidefinite Programming vs. LP Relaxations for Polynomial Programming," Mathematics of Operations Research, INFORMS, vol. 27(2), pages 347-360, May.
    7. Polyxeni-Margarita Kleniati & Panos Parpas & Berç Rustem, 2010. "Partitioning procedure for polynomial optimization," Journal of Global Optimization, Springer, vol. 48(4), pages 549-567, December.
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    Cited by:

    1. T. D. Chuong & V. Jeyakumar & G. Li, 2019. "A new bounded degree hierarchy with SOCP relaxations for global polynomial optimization and conic convex semi-algebraic programs," Journal of Global Optimization, Springer, vol. 75(4), pages 885-919, December.

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