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Comparison of Lasserre’s Measure-Based Bounds for Polynomial Optimization to Bounds Obtained by Simulated Annealing

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  • Etienne de Klerk

    (Tilburg University, 5000 LE Tilburg, Netherlands; and Delft University of Technology, 2600 AA Delft, Netherlands)

  • Monique Laurent

    (Tilburg University, 5000 LE Tilburg, Netherlands; and Centrum Wiskunde & Informatica, 1090 GB Amsterdam, Netherlands)

Abstract

We consider the problem of minimizing a continuous function f over a compact set K . We compare the hierarchy of upper bounds proposed by Lasserre [Lasserre JB (2011) A new look at nonnegativity on closed sets and polynomial optimization. SIAM J. Optim. 21(3):864–885] to bounds that may be obtained from simulated annealing. We show that, when f is a polynomial and K a convex body, this comparison yields a faster rate of convergence of the Lasserre hierarchy than what was previously known in the literature.

Suggested Citation

  • Etienne de Klerk & Monique Laurent, 2018. "Comparison of Lasserre’s Measure-Based Bounds for Polynomial Optimization to Bounds Obtained by Simulated Annealing," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1317-1325, November.
    • Handle: RePEc:inm:ormoor:v:43:y:2018:i:4:p:1317-1325
    DOI: 10.1287/moor.2017.0906
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    References listed on IDEAS

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    1. Etienne de Klerk & Jean B. Lasserre & Monique Laurent & Zhao Sun, 2017. "Bound-Constrained Polynomial Optimization Using Only Elementary Calculations," Mathematics of Operations Research, INFORMS, vol. 42(3), pages 834-853, August.
    2. Robert L. Smith, 1984. "Efficient Monte Carlo Procedures for Generating Points Uniformly Distributed over Bounded Regions," Operations Research, INFORMS, vol. 32(6), pages 1296-1308, December.
    3. Laurent, M., 2009. "Sums of squares, moment matrices and optimization over polynomials," Other publications TiSEM 9fef820b-69d2-43f2-a501-e, Tilburg University, School of Economics and Management.
    4. de Klerk, Etienne & Hess, Roxana & Laurent, Monique, 2017. "Improved convergence rates for Lasserre-type hierarchies of upper bounds for box-constrained polynomial optimization," Other publications TiSEM 66281fb7-02b8-4d87-930b-0, Tilburg University, School of Economics and Management.
    5. Adam Tauman Kalai & Santosh Vempala, 2006. "Simulated Annealing for Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 31(2), pages 253-266, May.
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    Cited by:

    1. Riley Badenbroek & Etienne Klerk, 2022. "Simulated Annealing for Convex Optimization: Rigorous Complexity Analysis and Practical Perspectives," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 465-491, August.
    2. de Klerk, Etienne & Badenbroek, Riley, 2022. "Simulated annealing with hit-and-run for convex optimization: complexity analysis and practical perspectives," Other publications TiSEM 323b4588-65e0-4889-a555-9, Tilburg University, School of Economics and Management.
    3. Etienne de Klerk & Monique Laurent, 2020. "Worst-Case Examples for Lasserre’s Measure–Based Hierarchy for Polynomial Optimization on the Hypercube," Mathematics of Operations Research, INFORMS, vol. 45(1), pages 86-98, February.

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