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Numerical certification of Pareto optimality for biobjective nonlinear problems

Author

Listed:
  • Charles Audet

    (Polytechnique Montréal)

  • Frédéric Messine

    (University of Toulouse, ENSEEIHT-LAPLACE)

  • Jordan Ninin

    (ENSTA-Bretagne, Lab-STICC, Team PRASYS)

Abstract

The solution to a biobjective optimization problem is composed of a collection of trade-off solution called the Pareto set. Based on a computer assisted proof methodology, the present work studies the question of certifying numerically that a conjectured set is close to the Pareto set. Two situations are considered. First, we analyze the case where the conjectured set is directly provided: one objective is explicitly given as a function of the other. Second, we analyze the situation where the conjectured set is parameterized: both objectives are explicitly given as functions of a parameter. In both cases, we formulate the question of verifying that the conjectured set is close to the Pareto set as a global optimization problem. These situations are illustrated on a new class of extremal problems over convex polygons in the plane. The objectives are to maximize the area and perimeter of a polygon with a fixed diameter, for a given number of sides.

Suggested Citation

  • Charles Audet & Frédéric Messine & Jordan Ninin, 2022. "Numerical certification of Pareto optimality for biobjective nonlinear problems," Journal of Global Optimization, Springer, vol. 83(4), pages 891-908, August.
    • Handle: RePEc:spr:jglopt:v:83:y:2022:i:4:d:10.1007_s10898-022-01127-1
    DOI: 10.1007/s10898-022-01127-1
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    References listed on IDEAS

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    1. Martin, Benjamin & Goldsztejn, Alexandre & Granvilliers, Laurent & Jermann, Christophe, 2017. "Constraint propagation using dominance in interval Branch & Bound for nonlinear biobjective optimization," European Journal of Operational Research, Elsevier, vol. 260(3), pages 934-948.
    2. Alexandre Goldsztejn & Ferenc Domes & Brice Chevalier, 2014. "First order rejection tests for multiple-objective optimization," Journal of Global Optimization, Springer, vol. 58(4), pages 653-672, April.
    3. Charles Audet & Pierre Hansen & Dragutin Svrtan, 2021. "Using symbolic calculations to determine largest small polygons," Journal of Global Optimization, Springer, vol. 81(1), pages 261-268, September.
    4. Benjamin Martin & Alexandre Goldsztejn & Laurent Granvilliers & Christophe Jermann, 2016. "On continuation methods for non-linear bi-objective optimization: towards a certified interval-based approach," Journal of Global Optimization, Springer, vol. 64(1), pages 3-16, January.
    5. Audet, Charles & Bigeon, Jean & Cartier, Dominique & Le Digabel, Sébastien & Salomon, Ludovic, 2021. "Performance indicators in multiobjective optimization," European Journal of Operational Research, Elsevier, vol. 292(2), pages 397-422.
    Full references (including those not matched with items on IDEAS)

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