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On upper approximations of Pareto fronts

Author

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  • I. Kaliszewski

    (Polish Academy of Sciences
    Warsaw School of Information Technology)

  • J. Miroforidis

    (Polish Academy of Sciences)

Abstract

In one of our earlier works, we proposed to approximate Pareto fronts to multiobjective optimization problems by two-sided approximations, one from inside and another from outside of the feasible objective set, called, respectively, lower shell and upper shell. We worked there under the assumption that for a given problem an upper shell exists. As it is not always the case, in this paper we give some sufficient conditions for the existence of upper shells. We also investigate how to constructively search infeasible sets to derive upper shells. We approach this issue by means of problem relaxations. We formally show that under certain conditions some subsets of lower shells to relaxed multiobjective optimization problems are upper shells in the respective unrelaxed problems. Results are illustrated by a numerical example representing a small but real mechanical problem. Practical implications of the results are discussed.

Suggested Citation

  • I. Kaliszewski & J. Miroforidis, 2018. "On upper approximations of Pareto fronts," Journal of Global Optimization, Springer, vol. 72(3), pages 475-490, November.
    • Handle: RePEc:spr:jglopt:v:72:y:2018:i:3:d:10.1007_s10898-018-0642-1
    DOI: 10.1007/s10898-018-0642-1
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    References listed on IDEAS

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    1. Ignacy Kaliszewski, 2006. "Soft Computing For Complex Multiple Criteria Decision Making," International Series in Operations Research and Management Science, Springer, number 978-0-387-30177-8, March.
    2. Kathrin Klamroth & Jørgen Tind & Margaret M. Wiecek, 2003. "Unbiased approximation in multicriteria optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 56(3), pages 413-437, January.
    3. Markus Hartikainen & Kaisa Miettinen & Margaret M. Wiecek, 2011. "Decision Making on Pareto Front Approximations with Inherent Nondominance," Lecture Notes in Economics and Mathematical Systems, in: Yong Shi & Shouyang Wang & Gang Kou & Jyrki Wallenius (ed.), New State of MCDM in the 21st Century, chapter 0, pages 35-45, Springer.
    4. Kaliszewski, Ignacy & Miroforidis, Janusz & Podkopaev, Dmitry, 2012. "Interactive Multiple Criteria Decision Making based on preference driven Evolutionary Multiobjective Optimization with controllable accuracy," European Journal of Operational Research, Elsevier, vol. 216(1), pages 188-199.
    5. Markus Hartikainen & Kaisa Miettinen & Margaret Wiecek, 2012. "PAINT: Pareto front interpolation for nonlinear multiobjective optimization," Computational Optimization and Applications, Springer, vol. 52(3), pages 845-867, July.
    6. I. Kaliszewski & J. Miroforidis, 2014. "Two-Sided Pareto Front Approximations," Journal of Optimization Theory and Applications, Springer, vol. 162(3), pages 845-855, September.
    7. Daniel Vanderpooten & Lakmali Weerasena & Margaret M. Wiecek, 2017. "Covers and approximations in multiobjective optimization," Journal of Global Optimization, Springer, vol. 67(3), pages 601-619, March.
    8. S. Ruzika & M. M. Wiecek, 2005. "Approximation Methods in Multiobjective Programming," Journal of Optimization Theory and Applications, Springer, vol. 126(3), pages 473-501, September.
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    Cited by:

    1. Janusz Miroforidis, 2021. "Bounds on efficient outcomes for large-scale cardinality-constrained Markowitz problems," Journal of Global Optimization, Springer, vol. 80(3), pages 617-634, July.
    2. Lakmali Weerasena, 2022. "Advancing local search approximations for multiobjective combinatorial optimization problems," Journal of Combinatorial Optimization, Springer, vol. 43(3), pages 589-612, April.
    3. I. Kaliszewski & J. Miroforidis, 2022. "Probing the Pareto front of a large-scale multiobjective problem with a MIP solver," Operational Research, Springer, vol. 22(5), pages 5617-5673, November.
    4. I. Kaliszewski & J. Miroforidis, 2021. "Cooperative multiobjective optimization with bounds on objective functions," Journal of Global Optimization, Springer, vol. 79(2), pages 369-385, February.

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