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Stochastic Optimization over a Pareto Set Associated with a Stochastic Multi-Objective Optimization Problem

Author

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  • Henri Bonnel

    (University of New Caledonia)

  • Julien Collonge

    (University of New Caledonia)

Abstract

We deal with the problem of minimizing the expectation of a real valued random function over the weakly Pareto or Pareto set associated with a Stochastic Multi-objective Optimization Problem, whose objectives are expectations of random functions. Assuming that the closed form of these expectations is difficult to obtain, we apply the Sample Average Approximation method in order to approach this problem. We prove that the Hausdorff–Pompeiu distance between the weakly Pareto sets associated with the Sample Average Approximation problem and the true weakly Pareto set converges to zero almost surely as the sample size goes to infinity, assuming that our Stochastic Multi-objective Optimization Problem is strictly convex. Then we show that every cluster point of any sequence of optimal solutions of the Sample Average Approximation problems is almost surely a true optimal solution. To handle also the non-convex case, we assume that the real objective to be minimized over the Pareto set depends on the expectations of the objectives of the Stochastic Optimization Problem, i.e. we optimize over the image space of the Stochastic Optimization Problem. Then, without any convexity hypothesis, we obtain the same type of results for the Pareto sets in the image spaces. Thus we show that the sequence of optimal values of the Sample Average Approximation problems converges almost surely to the true optimal value as the sample size goes to infinity.

Suggested Citation

  • Henri Bonnel & Julien Collonge, 2014. "Stochastic Optimization over a Pareto Set Associated with a Stochastic Multi-Objective Optimization Problem," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 405-427, August.
  • Handle: RePEc:spr:joptap:v:162:y:2014:i:2:d:10.1007_s10957-013-0367-8
    DOI: 10.1007/s10957-013-0367-8
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    References listed on IDEAS

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    1. Regina S. Burachik & Alfredo N. Iusem, 2008. "Set-Valued Mappings and Enlargements of Monotone Operators," Springer Optimization and Its Applications, Springer, number 978-0-387-69757-4, June.
    2. Horst, Reiner & Thoai, Nguyen V., 1999. "Maximizing a concave function over the efficient or weakly-efficient set," European Journal of Operational Research, Elsevier, vol. 117(2), pages 239-252, September.
    3. S. Dempe & N. Gadhi & A. B. Zemkoho, 2013. "New Optimality Conditions for the Semivectorial Bilevel Optimization Problem," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 54-74, April.
    4. H. P. Benson, 1998. "Further Analysis of an Outcome Set-Based Algorithm for Multiple-Objective Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 97(1), pages 1-10, April.
    5. Regina S. Burachik & Alfredo N. Iusem, 2008. "Enlargements of Monotone Operators," Springer Optimization and Its Applications, in: Set-Valued Mappings and Enlargements of Monotone Operators, chapter 0, pages 161-220, Springer.
    6. Jörg Fliege & Huifu Xu, 2011. "Stochastic Multiobjective Optimization: Sample Average Approximation and Applications," Journal of Optimization Theory and Applications, Springer, vol. 151(1), pages 135-162, October.
    7. Ankhili, Z. & Mansouri, A., 2009. "An exact penalty on bilevel programs with linear vector optimization lower level," European Journal of Operational Research, Elsevier, vol. 197(1), pages 36-41, August.
    8. H. P. Benson, 1998. "Hybrid Approach for Solving Multiple-Objective Linear Programs in Outcome Space," Journal of Optimization Theory and Applications, Springer, vol. 98(1), pages 17-35, July.
    9. H. Bonnel & J. Morgan, 2006. "Semivectorial Bilevel Optimization Problem: Penalty Approach," Journal of Optimization Theory and Applications, Springer, vol. 131(3), pages 365-382, December.
    10. R. Horst & N. V. Thoai & Y. Yamamoto & D. Zenke, 2007. "On Optimization over the Efficient Set in Linear Multicriteria Programming," Journal of Optimization Theory and Applications, Springer, vol. 134(3), pages 433-443, September.
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    Cited by:

    1. Fabrice Poirion & Quentin Mercier & Jean-Antoine Désidéri, 2017. "Descent algorithm for nonsmooth stochastic multiobjective optimization," Computational Optimization and Applications, Springer, vol. 68(2), pages 317-331, November.
    2. Henri Bonnel & Léonard Todjihoundé & Constantin Udrişte, 2015. "Semivectorial Bilevel Optimization on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 167(2), pages 464-486, November.
    3. Alexander Engau, 2017. "Proper Efficiency and Tradeoffs in Multiple Criteria and Stochastic Optimization," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 119-134, January.
    4. Hadi Karimi & Sandra D. Ekşioğlu & Michael Carbajales-Dale, 2021. "A biobjective chance constrained optimization model to evaluate the economic and environmental impacts of biopower supply chains," Annals of Operations Research, Springer, vol. 296(1), pages 95-130, January.
    5. Mercier, Quentin & Poirion, Fabrice & Désidéri, Jean-Antoine, 2018. "A stochastic multiple gradient descent algorithm," European Journal of Operational Research, Elsevier, vol. 271(3), pages 808-817.
    6. Henri Bonnel & Christopher Schneider, 2019. "Post-Pareto Analysis and a New Algorithm for the Optimal Parameter Tuning of the Elastic Net," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 993-1027, December.

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