IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v74y2019i3d10.1007_s10898-019-00745-6.html
   My bibliography  Save this article

Multi-objective unconstrained combinatorial optimization: a polynomial bound on the number of extreme supported solutions

Author

Listed:
  • Britta Schulze

    (University of Wuppertal)

  • Kathrin Klamroth

    (University of Wuppertal)

  • Michael Stiglmayr

    (University of Wuppertal)

Abstract

The multi-objective unconstrained combinatorial optimization problem (MUCO) can be considered as an archetype of a discrete linear multi-objective optimization problem. It can be interpreted as a specific relaxation of any multi-objective combinatorial optimization problem with linear sum objective function. While its single criteria analogon is analytically solvable, MUCO shares the computational complexity issues of most multi-objective combinatorial optimization problems: intractability and NP-hardness of the $$\varepsilon $$ ε -constraint scalarizations. In this article interrelations between the supported points of a MUCO problem, arrangements of hyperplanes and a weight space decomposition, and zonotopes are presented. Based on these interrelations and a result by Zaslavsky on the number of faces in an arrangement of hyperplanes, a polynomial bound on the number of extreme supported solutions can be derived, leading to an exact polynomial time algorithm to find all extreme supported solutions. It is shown how this algorithm can be incorporated into a solution approach for multi-objective knapsack problems.

Suggested Citation

  • Britta Schulze & Kathrin Klamroth & Michael Stiglmayr, 2019. "Multi-objective unconstrained combinatorial optimization: a polynomial bound on the number of extreme supported solutions," Journal of Global Optimization, Springer, vol. 74(3), pages 495-522, July.
    • Handle: RePEc:spr:jglopt:v:74:y:2019:i:3:d:10.1007_s10898-019-00745-6
    DOI: 10.1007/s10898-019-00745-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-019-00745-6
    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/s10898-019-00745-6?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. Ferrez, J.-A. & Fukuda, K. & Liebling, Th.M., 2005. "Solving the fixed rank convex quadratic maximization in binary variables by a parallel zonotope construction algorithm," European Journal of Operational Research, Elsevier, vol. 166(1), pages 35-50, October.
    2. Y. P. Aneja & K. P. K. Nair, 1979. "Bicriteria Transportation Problem," Management Science, INFORMS, vol. 25(1), pages 73-78, January.
    3. Saul Gass & Thomas Saaty, 1955. "The computational algorithm for the parametric objective function," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 2(1‐2), pages 39-45, March.
    4. Benson, Harold P. & Sun, Erjiang, 2002. "A weight set decomposition algorithm for finding all efficient extreme points in the outcome set of a multiple objective linear program," European Journal of Operational Research, Elsevier, vol. 139(1), pages 26-41, May.
    5. Hassene AISSI & A. Ridha MAHJOUB & S. Thomas McCORMICK & Maurice QUEYRANNE, 2015. "Strongly Polynomial Bounds for Multiobjective and Parametric Global minimum Cuts in Graphs and Hypergraphs," LIDAM Reprints CORE 2751, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Anthony Przybylski & Xavier Gandibleux & Matthias Ehrgott, 2010. "A Recursive Algorithm for Finding All Nondominated Extreme Points in the Outcome Set of a Multiobjective Integer Programme," INFORMS Journal on Computing, INFORMS, vol. 22(3), pages 371-386, August.
    7. HASSENE, Aissi & MAHJOUB, A. Ridha & McCORMICK, S. Thomas & QUEYRANNE, Maurice, 2015. "Strongly polynomial bounds for multiobjective and parametric global minimum cuts in graphs and hypergraphs," LIDAM Discussion Papers CORE 2015004, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    8. Matthias Ehrgott & Andreas Löhne & Lizhen Shao, 2012. "A dual variant of Benson’s “outer approximation algorithm” for multiple objective linear programming," Journal of Global Optimization, Springer, vol. 52(4), pages 757-778, April.
    9. David Pisinger, 1997. "A Minimal Algorithm for the 0-1 Knapsack Problem," Operations Research, INFORMS, vol. 45(5), pages 758-767, October.
    10. H. P. Benson & E. Sun, 2000. "Outcome Space Partition of the Weight Set in Multiobjective Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 105(1), pages 17-36, April.
    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. Stephan Helfrich & Kathrin Prinz & Stefan Ruzika, 2024. "The Weighted p-Norm Weight Set Decomposition for Multiobjective Discrete Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 202(3), pages 1187-1216, September.
    2. Stephan Helfrich & Tyler Perini & Pascal Halffmann & Natashia Boland & Stefan Ruzika, 2023. "Analysis of the weighted Tchebycheff weight set decomposition for multiobjective discrete optimization problems," Journal of Global Optimization, Springer, vol. 86(2), pages 417-440, June.

    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. Özgür Özpeynirci & Murat Köksalan, 2010. "An Exact Algorithm for Finding Extreme Supported Nondominated Points of Multiobjective Mixed Integer Programs," Management Science, INFORMS, vol. 56(12), pages 2302-2315, December.
    2. Stephan Helfrich & Kathrin Prinz & Stefan Ruzika, 2024. "The Weighted p-Norm Weight Set Decomposition for Multiobjective Discrete Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 202(3), pages 1187-1216, September.
    3. Melih Ozlen & Benjamin A. Burton & Cameron A. G. MacRae, 2014. "Multi-Objective Integer Programming: An Improved Recursive Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 470-482, February.
    4. Anthony Przybylski & Xavier Gandibleux & Matthias Ehrgott, 2010. "A Recursive Algorithm for Finding All Nondominated Extreme Points in the Outcome Set of a Multiobjective Integer Programme," INFORMS Journal on Computing, INFORMS, vol. 22(3), pages 371-386, August.
    5. Fritz Bökler & Sophie N. Parragh & Markus Sinnl & Fabien Tricoire, 2024. "An outer approximation algorithm for generating the Edgeworth–Pareto hull of multi-objective mixed-integer linear programming problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(1), pages 263-290, August.
    6. Stephan Helfrich & Tyler Perini & Pascal Halffmann & Natashia Boland & Stefan Ruzika, 2023. "Analysis of the weighted Tchebycheff weight set decomposition for multiobjective discrete optimization problems," Journal of Global Optimization, Springer, vol. 86(2), pages 417-440, June.
    7. Piercy, Craig A. & Steuer, Ralph E., 2019. "Reducing wall-clock time for the computation of all efficient extreme points in multiple objective linear programming," European Journal of Operational Research, Elsevier, vol. 277(2), pages 653-666.
    8. Przybylski, Anthony & Gandibleux, Xavier, 2017. "Multi-objective branch and bound," European Journal of Operational Research, Elsevier, vol. 260(3), pages 856-872.
    9. Julius Bauß & Michael Stiglmayr, 2024. "Augmenting bi-objective branch and bound by scalarization-based information," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(1), pages 85-121, August.
    10. Stephan Helfrich & Arne Herzel & Stefan Ruzika & Clemens Thielen, 2022. "An approximation algorithm for a general class of multi-parametric optimization problems," Journal of Combinatorial Optimization, Springer, vol. 44(3), pages 1459-1494, October.
    11. Alves, Maria João & Costa, João Paulo, 2016. "Graphical exploration of the weight space in three-objective mixed integer linear programs," European Journal of Operational Research, Elsevier, vol. 248(1), pages 72-83.
    12. José Figueira & Luís Paquete & Marco Simões & Daniel Vanderpooten, 2013. "Algorithmic improvements on dynamic programming for the bi-objective {0,1} knapsack problem," Computational Optimization and Applications, Springer, vol. 56(1), pages 97-111, September.
    13. Klamroth, Kathrin & Stiglmayr, Michael & Sudhoff, Julia, 2023. "Ordinal optimization through multi-objective reformulation," European Journal of Operational Research, Elsevier, vol. 311(2), pages 427-443.
    14. Cristina Bazgan & Arne Herzel & Stefan Ruzika & Clemens Thielen & Daniel Vanderpooten, 2024. "Approximating multiobjective optimization problems: How exact can you be?," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(1), pages 5-25, August.
    15. Pascal Halffmann & Tobias Dietz & Anthony Przybylski & Stefan Ruzika, 2020. "An inner approximation method to compute the weight set decomposition of a triobjective mixed-integer problem," Journal of Global Optimization, Springer, vol. 77(4), pages 715-742, August.
    16. Özgür Özpeynirci, 2017. "On nadir points of multiobjective integer programming problems," Journal of Global Optimization, Springer, vol. 69(3), pages 699-712, November.
    17. Soylu, Banu, 2018. "The search-and-remove algorithm for biobjective mixed-integer linear programming problems," European Journal of Operational Research, Elsevier, vol. 268(1), pages 281-299.
    18. Daniel Jornada & V. Jorge Leon, 2020. "Filtering Algorithms for Biobjective Mixed Binary Linear Optimization Problems with a Multiple-Choice Constraint," INFORMS Journal on Computing, INFORMS, vol. 32(1), pages 57-73, January.
    19. Natashia Boland & Hadi Charkhgard & Martin Savelsbergh, 2015. "A Criterion Space Search Algorithm for Biobjective Integer Programming: The Balanced Box Method," INFORMS Journal on Computing, INFORMS, vol. 27(4), pages 735-754, November.
    20. S. Dutta & S. Acharya & Rajashree Mishra, 2016. "Genetic algorithm based fuzzy stochastic transportation programming problem with continuous random variables," OPSEARCH, Springer;Operational Research Society of India, vol. 53(4), pages 835-872, December.

    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:74:y:2019:i:3:d:10.1007_s10898-019-00745-6. 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.