IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v83y2022i4d10.1007_s10898-022-01127-1.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-022-01127-1
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10898-022-01127-1?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    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. 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.
    4. 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.
    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)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ignacio Araya & Jose Campusano & Damir Aliquintui, 2019. "Nonlinear biobjective optimization: improvements to interval branch & bound algorithms," Journal of Global Optimization, Springer, vol. 75(1), pages 91-110, September.
    2. Ignacio Araya & Damir Aliquintui & Franco Ardiles & Braulio Lobo, 2021. "Nonlinear biobjective optimization: improving the upper envelope using feasible line segments," Journal of Global Optimization, Springer, vol. 79(2), pages 503-520, February.
    3. 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.
    4. Marendet, Antoine & Goldsztejn, Alexandre & Chabert, Gilles & Jermann, Christophe, 2020. "A standard branch-and-bound approach for nonlinear semi-infinite problems," European Journal of Operational Research, Elsevier, vol. 282(2), pages 438-452.
    5. Zandieh, Fatemeh & Ghannadpour, Seyed Farid, 2023. "A comprehensive risk assessment view on interval type-2 fuzzy controller for a time-dependent HazMat routing problem," European Journal of Operational Research, Elsevier, vol. 305(2), pages 685-707.
    6. Dalila B. M. M. Fontes & S. Mahdi Homayouni, 2023. "A bi-objective multi-population biased random key genetic algorithm for joint scheduling quay cranes and speed adjustable vehicles in container terminals," Flexible Services and Manufacturing Journal, Springer, vol. 35(1), pages 241-268, March.
    7. Lourdes Uribe & Johan M Bogoya & Andrés Vargas & Adriana Lara & Günter Rudolph & Oliver Schütze, 2020. "A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems," Mathematics, MDPI, vol. 8(10), pages 1-29, October.
    8. Mesquita-Cunha, Mariana & Figueira, José Rui & Barbosa-Póvoa, Ana Paula, 2023. "New ϵ−constraint methods for multi-objective integer linear programming: A Pareto front representation approach," European Journal of Operational Research, Elsevier, vol. 306(1), pages 286-307.
    9. Gaggero, Mauro & Paolucci, Massimo & Ronco, Roberto, 2023. "Exact and heuristic solution approaches for energy-efficient identical parallel machine scheduling with time-of-use costs," European Journal of Operational Research, Elsevier, vol. 311(3), pages 845-866.
    10. Ducardo L. Molina & Juan Ricardo Vidal Medina & Alexis Sagastume Gutiérrez & Juan J. Cabello Eras & Jesús A. Lopez & Simón Hincapie & Enrique C. Quispe, 2023. "Multiobjective Optimization of the Energy Efficiency and the Steam Flow in a Bagasse Boiler," Sustainability, MDPI, vol. 15(14), pages 1-17, July.
    11. Jean Bigeon & Sébastien Le Digabel & Ludovic Salomon, 2021. "DMulti-MADS: mesh adaptive direct multisearch for bound-constrained blackbox multiobjective optimization," Computational Optimization and Applications, Springer, vol. 79(2), pages 301-338, June.
    12. Francisco Jonatas Siqueira Coelho & Allan Rivalles Souza Feitosa & André Luís Michels Alcântara & Kaifeng Li & Ronaldo Ferreira Lima & Victor Rios Silva & Abel Guilhermino da Silva-Filho, 2023. "HyMOTree: Automatic Hyperparameters Tuning for Non-Technical Loss Detection Based on Multi-Objective and Tree-Based Algorithms," Energies, MDPI, vol. 16(13), pages 1-22, June.
    13. Li, Mingwu & Dankowicz, Harry, 2020. "Optimization with equality and inequality constraints using parameter continuation," Applied Mathematics and Computation, Elsevier, vol. 375(C).
    14. Gholamreza Shojatalab & Seyed Hadi Nasseri & Iraj Mahdavi, 2023. "New multi-objective optimization model for tourism systems with fuzzy data and new approach developed epsilon constraint method," OPSEARCH, Springer;Operational Research Society of India, vol. 60(3), pages 1360-1385, September.
    15. Ricardo Landa & Giomara Lárraga & Gregorio Toscano, 2019. "Use of a goal-constraint-based approach for finding the region of interest in multi-objective problems," Journal of Heuristics, Springer, vol. 25(1), pages 107-139, February.
    16. Tiago Montanher & Arnold Neumaier & Ferenc Domes, 2018. "A computational study of global optimization solvers on two trust region subproblems," Journal of Global Optimization, Springer, vol. 71(4), pages 915-934, August.
    17. Raka Jovanovic & Antonio P. Sanfilippo & Stefan Voß, 2022. "Fixed set search applied to the multi-objective minimum weighted vertex cover problem," Journal of Heuristics, Springer, vol. 28(4), pages 481-508, August.
    18. Rohit Kannan & Paul I. Barton, 2017. "The cluster problem in constrained global optimization," Journal of Global Optimization, Springer, vol. 69(3), pages 629-676, November.
    19. Gonzalo Sánchez-Contreras & Adrián Fernández-Rodríguez & Antonio Fernández-Cardador & Asunción P. Cucala, 2023. "A Two-Level Fuzzy Multi-Objective Design of ATO Driving Commands for Energy-Efficient Operation of Metropolitan Railway Lines," Sustainability, MDPI, vol. 15(12), pages 1-24, June.
    20. Duro, João A. & Ozturk, Umud Esat & Oara, Daniel C. & Salomon, Shaul & Lygoe, Robert J. & Burke, Richard & Purshouse, Robin C., 2023. "Methods for constrained optimization of expensive mixed-integer multi-objective problems, with application to an internal combustion engine design problem," European Journal of Operational Research, Elsevier, vol. 307(1), pages 421-446.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jglopt:v:83:y:2022:i:4:d:10.1007_s10898-022-01127-1. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.