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Alternative SDP and SOCP approximations for polynomial optimization

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.

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Acknowledgements

The authors would like to thank two anonymous referees and an Associate Editor for their constructive and insightful comments which greatly improved this article. The work of the first and last authors was supported by NSF Grant CMMI-1300193. Bissan Ghaddar was supported by NSERC Discovery Grant RGPIN-2017-04185, and Joe Naoum-Sawaya was supported by NSERC Discovery Grant RGPIN-2017-03962.

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Correspondence to Xiaolong Kuang.

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Kuang, X., Ghaddar, B., Naoum-Sawaya, J. et al. Alternative SDP and SOCP approximations for polynomial optimization. EURO J Comput Optim 7, 153–175 (2019). https://doi.org/10.1007/s13675-018-0101-2

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  • DOI: https://doi.org/10.1007/s13675-018-0101-2

Keywords

  • Polynomial optimization
  • Second-order cone relaxation
  • Semidefinite relaxation
  • Approximation hierarchy

Mathematics Subject Classification

  • 90C22
  • 90C26