IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v108y2001i1d10.1023_a1026465906067.html
   My bibliography  Save this article

Duality of Nonscalarized Multiobjective Linear Programs: Dual Balance, Level Sets, and Dual Clusters of Optimal Vectors

Author

Listed:
  • E. Galperin

    (Université du Québec à Montréal)

  • P. Jimenez Guerra

    (Universidad Nacional de Educación a Distancia)

Abstract

A new concept of duality is proposed for multiobjective linear programs. It is based on a set expansion process for the computation of optimal solutions without scalarization. The duality gap qualifications are investigated; the primal–dual balance set and level set equations are derived. It is demonstrated that the nonscalarized dual problem presents a cluster of optimal dual vectors that corresponds to a unique optimal primal vector. Comparisons are made with linear utility, minmax and minmin scalarizations. Connections to Pareto optimality are studied and relations to sensitivity and parametric programming are discussed. The ideas are illustrated by examples.

Suggested Citation

  • E. Galperin & P. Jimenez Guerra, 2001. "Duality of Nonscalarized Multiobjective Linear Programs: Dual Balance, Level Sets, and Dual Clusters of Optimal Vectors," Journal of Optimization Theory and Applications, Springer, vol. 108(1), pages 109-137, January.
  • Handle: RePEc:spr:joptap:v:108:y:2001:i:1:d:10.1023_a:1026465906067
    DOI: 10.1023/A:1026465906067
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/A:1026465906067
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1023/A:1026465906067?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. E. A. Galperin, 1997. "Pareto Analysis vis-à-vis Balance Space Approach in Multiobjective Global Optimization," Journal of Optimization Theory and Applications, Springer, vol. 93(3), pages 533-545, June.
    2. M. Ehrgott & H. W. Hamacher & K. Klamroth & S. Nickel & A. Schöbel & M. M. Wiecek, 1997. "Equivalence of Balance Points and Pareto Solutions in Multiple-Objective Programming," Journal of Optimization Theory and Applications, Springer, vol. 92(1), pages 209-212, January.
    3. Ben-Israel, A & Ben-Tal, A & Charnes, A, 1977. "Necessary and Sufficient Conditions for a Pareto Optimum in Convex Programming," Econometrica, Econometric Society, vol. 45(4), pages 811-820, May.
    4. Balbas, Alejandro & Heras, Antonio, 1993. "Duality theory for infinite-dimensional multiobjective linear programming," European Journal of Operational Research, Elsevier, vol. 68(3), pages 379-388, August.
    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. N. Mahdavi-Amiri & F. Salehi Sadaghiani, 2017. "Strictly feasible solutions and strict complementarity in multiple objective linear optimization," 4OR, Springer, vol. 15(3), pages 303-326, September.
    2. A. Balbás & E. Galperin & P. Jiménez-Guerra, 2002. "Radial Solutions and Orthogonal Trajectories in Multiobjective Global Optimization," Journal of Optimization Theory and Applications, Springer, vol. 115(2), pages 315-344, November.
    3. Luc, Dinh The, 2011. "On duality in multiple objective linear programming," European Journal of Operational Research, Elsevier, vol. 210(2), pages 158-168, 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. A. Balbás & E. Galperin & P. Jiménez-Guerra, 2002. "Radial Solutions and Orthogonal Trajectories in Multiobjective Global Optimization," Journal of Optimization Theory and Applications, Springer, vol. 115(2), pages 315-344, November.
    2. Wassila Drici & Fatma Zohra Ouail & Mustapha Moulaï, 2018. "Optimizing a linear fractional function over the integer efficient set," Annals of Operations Research, Springer, vol. 267(1), pages 135-151, August.
    3. Jiménez Guerra, Pedro, 2006. "Generalized vector risk functions," DEE - Working Papers. Business Economics. WB wb066721, Universidad Carlos III de Madrid. Departamento de Economía de la Empresa.
    4. Luc, Dinh The, 2011. "On duality in multiple objective linear programming," European Journal of Operational Research, Elsevier, vol. 210(2), pages 158-168, April.
    5. Balbás, Alejandro & Balbás, Raquel & Mayoral, Silvia, 2009. "Portfolio choice and optimal hedging with general risk functions: A simplex-like algorithm," European Journal of Operational Research, Elsevier, vol. 192(2), pages 603-620, January.
    6. N. Mahdavi-Amiri & F. Salehi Sadaghiani, 2017. "Strictly feasible solutions and strict complementarity in multiple objective linear optimization," 4OR, Springer, vol. 15(3), pages 303-326, September.
    7. Hernández-Lerma, Onésimo & Romera, Rosario, 2000. "Pareto optimality in multiobjective Markov control processes," DES - Working Papers. Statistics and Econometrics. WS 9865, Universidad Carlos III de Madrid. Departamento de Estadística.
    8. E. A. Galperin, 1997. "Pareto Analysis vis-à-vis Balance Space Approach in Multiobjective Global Optimization," Journal of Optimization Theory and Applications, Springer, vol. 93(3), pages 533-545, June.
    9. Zerdani, Ouiza & Moulai, Mustapha, 2011. "Optimization over an integer efficient set of a Multiple Objective Linear Fractional Problem," MPRA Paper 35579, University Library of Munich, Germany.
    10. Muñoz-Bouzo, María José, 1997. "Stochastic measures of financial markets efficiency and integration," DEE - Working Papers. Business Economics. WB 7018, Universidad Carlos III de Madrid. Departamento de Economía de la Empresa.

    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:108:y:2001:i:1:d:10.1023_a:1026465906067. 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.