IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v203y2024i1d10.1007_s10957-024-02478-3.html
   My bibliography  Save this article

On Necessary Optimality Conditions for Sets of Points in Multiobjective Optimization

Author

Listed:
  • Andrea Cristofari

    (University of Rome “Tor Vergata”)

  • Marianna Santis

    (Sapienza University of Rome)

  • Stefano Lucidi

    (Sapienza University of Rome)

Abstract

Taking inspiration from what is commonly done in single-objective optimization, most local algorithms proposed for multiobjective optimization extend the classical iterative scalar methods and produce sequences of points able to converge to single efficient points. Recently, a growing number of local algorithms that build sequences of sets has been devised, following the real nature of multiobjective optimization, where the aim is that of approximating the efficient set. This calls for a new analysis of the necessary optimality conditions for multiobjective optimization. We explore conditions for sets of points that share the same features of the necessary optimality conditions for single-objective optimization. On the one hand, from a theoretical point of view, these conditions define properties that are necessarily satisfied by the (weakly) efficient set. On the other hand, from an algorithmic point of view, any set that does not satisfy the proposed conditions can be easily improved by using first-order information on some objective functions. We analyse both the unconstrained and the constrained case, giving some examples.

Suggested Citation

  • Andrea Cristofari & Marianna Santis & Stefano Lucidi, 2024. "On Necessary Optimality Conditions for Sets of Points in Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 126-145, October.
    • Handle: RePEc:spr:joptap:v:203:y:2024:i:1:d:10.1007_s10957-024-02478-3
    DOI: 10.1007/s10957-024-02478-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-024-02478-3
    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/s10957-024-02478-3?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. M. L. N. Gonçalves & F. S. Lima & L. F. Prudente, 2022. "Globally convergent Newton-type methods for multiobjective optimization," Computational Optimization and Applications, Springer, vol. 83(2), pages 403-434, November.
    2. Gabriele Eichfelder & Peter Kirst & Laura Meng & Oliver Stein, 2021. "Correction to: A general branch-and-bound framework for continuous global multiobjective optimization," Journal of Global Optimization, Springer, vol. 80(1), pages 229-229, May.
    3. Matteo Lapucci & Pierluigi Mansueto, 2023. "A limited memory Quasi-Newton approach for multi-objective optimization," Computational Optimization and Applications, Springer, vol. 85(1), pages 33-73, May.
    4. G. Cocchi & M. Lapucci, 2020. "An augmented Lagrangian algorithm for multi-objective optimization," Computational Optimization and Applications, Springer, vol. 77(1), pages 29-56, September.
    5. G. Cocchi & G. Liuzzi & S. Lucidi & M. Sciandrone, 2020. "On the convergence of steepest descent methods for multiobjective optimization," Computational Optimization and Applications, Springer, vol. 77(1), pages 1-27, September.
    6. Gabriele Eichfelder & Leo Warnow, 2022. "An approximation algorithm for multi-objective optimization problems using a box-coverage," Journal of Global Optimization, Springer, vol. 83(2), pages 329-357, June.
    7. Jörg Fliege & Benar Fux Svaiter, 2000. "Steepest descent methods for multicriteria optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 51(3), pages 479-494, August.
    8. Oliver Stein & Maximilian Volk, 2023. "Generalized Polarity and Weakest Constraint Qualifications in Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 198(3), pages 1156-1190, September.
    9. M. C. Maciel & S. A. Santos & G. N. Sottosanto, 2009. "Regularity Conditions in Differentiable Vector Optimization Revisited," Journal of Optimization Theory and Applications, Springer, vol. 142(2), pages 385-398, August.
    10. Gabriele Eichfelder & Peter Kirst & Laura Meng & Oliver Stein, 2021. "A general branch-and-bound framework for continuous global multiobjective optimization," Journal of Global Optimization, Springer, vol. 80(1), pages 195-227, May.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Pengyan Huang & Guangchen Wang & Shujun Wang, 2025. "Pareto Game of Stochastic Differential System with Terminal State Constraint," Journal of Optimization Theory and Applications, Springer, vol. 205(1), pages 1-30, April.

    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. Matteo Lapucci & Pierluigi Mansueto, 2023. "A limited memory Quasi-Newton approach for multi-objective optimization," Computational Optimization and Applications, Springer, vol. 85(1), pages 33-73, May.
    2. Marianna Santis & Gabriele Eichfelder & Daniele Patria & Leo Warnow, 2025. "Using dual relaxations in multiobjective mixed-integer convex quadratic programming," Journal of Global Optimization, Springer, vol. 92(1), pages 159-186, May.
    3. Ignacio Araya & Victor Reyes & Javier Montero, 2025. "Extending interval branch-and-bound from two to few objectives in nonlinear multiobjective optimization," Journal of Global Optimization, Springer, vol. 92(2), pages 295-320, June.
    4. G. A. Carrizo & N. S. Fazzio & M. D. Sánchez & M. L. Schuverdt, 2024. "Scaled-PAKKT sequential optimality condition for multiobjective problems and its application to an Augmented Lagrangian method," Computational Optimization and Applications, Springer, vol. 89(3), pages 769-803, December.
    5. Gabriele Eichfelder & Oliver Stein & Leo Warnow, 2024. "A Solver for Multiobjective Mixed-Integer Convex and Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1736-1766, November.
    6. Matteo Lapucci & Pierluigi Mansueto, 2024. "Cardinality-Constrained Multi-objective Optimization: Novel Optimality Conditions and Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 201(1), pages 323-351, April.
    7. Eichfelder, Gabriele & Warnow, Leo, 2023. "Advancements in the computation of enclosures for multi-objective optimization problems," European Journal of Operational Research, Elsevier, vol. 310(1), pages 315-327.
    8. Yong Zhao & Wang Chen & Xinmin Yang, 2024. "Adaptive Sampling Stochastic Multigradient Algorithm for Stochastic Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 215-241, January.
    9. Jamilu Yahaya & Poom Kumam & Mahmoud Muhammad Yahaya, 2025. "A New Hybrid Conjugate Gradient Method Based on a Convex Combination for Multiobjective Optimization," SN Operations Research Forum, Springer, vol. 6(2), pages 1-26, June.
    10. Luca Benvenuti & Alberto Santis & Marianna Santis & Daniele Patria, 2024. "Designing sustainable diet plans by solving triobjective integer programs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(3), pages 703-721, December.
    11. Gabriele Eichfelder & Leo Warnow, 2022. "An approximation algorithm for multi-objective optimization problems using a box-coverage," Journal of Global Optimization, Springer, vol. 83(2), pages 329-357, June.
    12. Moritz Link & Stefan Volkwein, 2023. "Adaptive piecewise linear relaxations for enclosure computations for nonconvex multiobjective mixed-integer quadratically constrained programs," Journal of Global Optimization, Springer, vol. 87(1), pages 97-132, September.
    13. Hiroki Tanabe & Ellen H. Fukuda & Nobuo Yamashita, 2023. "An accelerated proximal gradient method for multiobjective optimization," Computational Optimization and Applications, Springer, vol. 86(2), pages 421-455, November.
    14. Wei-tian Wu & Xin-min Yang, 2024. "Reference-point-based branch and bound algorithm for multiobjective optimization," Journal of Global Optimization, Springer, vol. 88(4), pages 927-945, April.
    15. Qing-Rui He & Chun-Rong Chen & Sheng-Jie Li, 2023. "Spectral conjugate gradient methods for vector optimization problems," Computational Optimization and Applications, Springer, vol. 86(2), pages 457-489, November.
    16. Gabriele Eichfelder & Leo Warnow, 2024. "A hybrid patch decomposition approach to compute an enclosure for multi-objective mixed-integer convex optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(1), pages 291-320, August.
    17. Qing-Rui He & Sheng-Jie Li & Bo-Ya Zhang & Chun-Rong Chen, 2024. "A family of conjugate gradient methods with guaranteed positiveness and descent for vector optimization," Computational Optimization and Applications, Springer, vol. 89(3), pages 805-842, December.
    18. Vahid Mahmoodian & Iman Dayarian & Payman Ghasemi Saghand & Yu Zhang & Hadi Charkhgard, 2022. "A Criterion Space Branch-and-Cut Algorithm for Mixed Integer Bilinear Maximum Multiplicative Programs," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1453-1470, May.
    19. Mehri Rashidi & Majid Soleimani-damaneh, 2024. "MultiSQP-GS: a sequential quadratic programming algorithm via gradient sampling for nonsmooth constrained multiobjective optimization," Computational Optimization and Applications, Springer, vol. 89(3), pages 729-767, December.
    20. Samira Fallah & Ted K. Ralphs & Natashia L. Boland, 2024. "On the relationship between the value function and the efficient frontier of a mixed integer linear optimization problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(1), pages 175-220, August.

    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:joptap:v:203:y:2024:i:1:d:10.1007_s10957-024-02478-3. 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.