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An Algorithm for the Global Optimization of a Class of Continuous Minimax Problems

Author

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  • P. Parpas

    (Imperial College)

  • B. Rustem

    (Imperial College)

Abstract

We propose an algorithm for the global optimization of continuous minimax problems involving polynomials. The method can be described as a discretization approach to the well known semi-infinite formulation of the problem. We proceed by approximating the infinite number of constraints using tools and techniques from semidefinite programming. We then show that, under appropriate conditions, the SDP approximation converges to the globally optimal solution of the problem. We also discuss the numerical performance of the method on some test problems.

Suggested Citation

  • P. Parpas & B. Rustem, 2009. "An Algorithm for the Global Optimization of a Class of Continuous Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 141(2), pages 461-473, May.
    • Handle: RePEc:spr:joptap:v:141:y:2009:i:2:d:10.1007_s10957-008-9473-4
    DOI: 10.1007/s10957-008-9473-4
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    References listed on IDEAS

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    1. Stanislav Žaković & Costas Pantelides & Berc Rustem, 2000. "An Interior Point Algorithm for Computing Saddle Points of Constrained Continuous Minimax," Annals of Operations Research, Springer, vol. 99(1), pages 59-77, December.
    2. Stanislav Žaković & Berc Rustem, 2003. "Semi-Infinite Programming and Applications to Minimax Problems," Annals of Operations Research, Springer, vol. 124(1), pages 81-110, November.
    3. B. Rustem & S. Žaković & P. Parpas, 2008. "Convergence of an Interior Point Algorithm for Continuous Minimax," Journal of Optimization Theory and Applications, Springer, vol. 136(1), pages 87-103, January.
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    Cited by:

    1. Julien Marzat & Eric Walter & Hélène Piet-Lahanier, 2013. "Worst-case global optimization of black-box functions through Kriging and relaxation," Journal of Global Optimization, Springer, vol. 55(4), pages 707-727, April.
    2. Feng Guo & Xiaoxia Sun, 2020. "On semi-infinite systems of convex polynomial inequalities and polynomial optimization problems," Computational Optimization and Applications, Springer, vol. 75(3), pages 669-699, April.
    3. J. Lasserre, 2011. "Min-max and robust polynomial optimization," Journal of Global Optimization, Springer, vol. 51(1), pages 1-10, September.
    4. Olga Kostyukova & Tatiana Tchemisova, 2017. "Optimality Conditions for Convex Semi-infinite Programming Problems with Finitely Representable Compact Index Sets," Journal of Optimization Theory and Applications, Springer, vol. 175(1), pages 76-103, October.
    5. Yu, Shui & Wu, Xiao & Zhao, Dongyu & Li, Yun, 2024. "A two-level surrogate framework for demand-objective time-variant reliability-based design optimization," Reliability Engineering and System Safety, Elsevier, vol. 244(C).
    6. Luo, Xin & Sun, Min, 2022. "Development of modal interval algorithm for solving continuous minimax problems," Applied Mathematics and Computation, Elsevier, vol. 422(C).
    7. Feng Guo & Liguo Jiao, 2021. "On solving a class of fractional semi-infinite polynomial programming problems," Computational Optimization and Applications, Springer, vol. 80(2), pages 439-481, November.
    8. Li Wang & Feng Guo, 2014. "Semidefinite relaxations for semi-infinite polynomial programming," Computational Optimization and Applications, Springer, vol. 58(1), pages 133-159, May.
    9. J. Lasserre, 2012. "An algorithm for semi-infinite polynomial optimization," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(1), pages 119-129, April.

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