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A multilevel analysis of the Lasserre hierarchy

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  • Campos, Juan S.
  • Misener, Ruth
  • Parpas, Panos

Abstract

This paper analyzes the relation between different orders of the Lasserre hierarchy for polynomial optimization (POP). Although for some cases solving the semidefinite programming relaxation corresponding to the first order of the hierarchy is enough to solve the underlying POP, other problems require sequentially solving the second or higher orders until a solution is found. For these cases, and assuming that the lower order semidefinite programming relaxation has been solved, we develop prolongation operators that exploit the solutions already calculated to find initial approximations for the solution of the higher order relaxation. We can prove feasibility in the higher order of the hierarchy of the points obtained using the operators, as well as convergence to the optimal as the relaxation order increases. Furthermore, the operators are simple and inexpensive for problems where the projection over the feasible set is “easy” to calculate (for example integer {0, 1} and {−1,1} POPs). Our numerical experiments show that it is possible to extract useful information for real applications using the prolongation operators. In particular, we illustrate how the operators can be used to increase the efficiency of an infeasible interior point method by using them as an initial point. We use this technique to solve quadratic integer {0, 1} problems, as well as MAX-CUT and integer partition problems.

Suggested Citation

  • Campos, Juan S. & Misener, Ruth & Parpas, Panos, 2019. "A multilevel analysis of the Lasserre hierarchy," European Journal of Operational Research, Elsevier, vol. 277(1), pages 32-41.
  • Handle: RePEc:eee:ejores:v:277:y:2019:i:1:p:32-41
    DOI: 10.1016/j.ejor.2019.02.016
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    References listed on IDEAS

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    1. Klerk, Etienne de, 2010. "Exploiting special structure in semidefinite programming: A survey of theory and applications," European Journal of Operational Research, Elsevier, vol. 201(1), pages 1-10, February.
    2. 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.
    3. Boukouvala, Fani & Misener, Ruth & Floudas, Christodoulos A., 2016. "Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO," European Journal of Operational Research, Elsevier, vol. 252(3), pages 701-727.
    4. Caprara, Alberto, 2008. "Constrained 0-1 quadratic programming: Basic approaches and extensions," European Journal of Operational Research, Elsevier, vol. 187(3), pages 1494-1503, June.
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