IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v132y2007i3d10.1007_s10957-007-9164-6.html
   My bibliography  Save this article

On Linear Programming Duality and Necessary and Sufficient Conditions in Minimax Theory

Author

Listed:
  • J. B. G. Frenk

    (Erasmus University)

  • P. Kas

    (Eastern Mediterranean University)

  • G. Kassay

    (Babes-Bolyai University)

Abstract

In this paper we discuss necessary and sufficient conditions for different minimax results to hold using only linear programming duality and the finite intersection property for compact sets. It turns out that these necessary and sufficient conditions have a clear interpretation within zero-sum game theory. We apply these results to derive necessary and sufficient conditions for strong duality for a general class of optimization problems.

Suggested Citation

  • J. B. G. Frenk & P. Kas & G. Kassay, 2007. "On Linear Programming Duality and Necessary and Sufficient Conditions in Minimax Theory," Journal of Optimization Theory and Applications, Springer, vol. 132(3), pages 423-439, March.
  • Handle: RePEc:spr:joptap:v:132:y:2007:i:3:d:10.1007_s10957-007-9164-6
    DOI: 10.1007/s10957-007-9164-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-007-9164-6
    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/s10957-007-9164-6?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. J. B. G. Frenk & G. Kassay, 1999. "On Classes of Generalized Convex Functions, Gordan–Farkas Type Theorems, and Lagrangian Duality," Journal of Optimization Theory and Applications, Springer, vol. 102(2), pages 315-343, August.
    2. Frenk, J.B.G. & Kas, P. & Kassay, G., 2004. "On linear programming duality and necessary and sufficient conditions in minimax theory," Econometric Institute Research Papers EI 2004-14, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    3. Frenk, J. B. G. & Kassay, G. & Kolumban, J., 2004. "On equivalent results in minimax theory," European Journal of Operational Research, Elsevier, vol. 157(1), pages 46-58, August.
    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. Frenk, J.B.G. & Kassay, G., 2005. "Lagrangian duality and cone convexlike functions," ERIM Report Series Research in Management ERS-2005-019-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    2. J. B. G. Frenk & G. Kassay, 2007. "Lagrangian Duality and Cone Convexlike Functions," Journal of Optimization Theory and Applications, Springer, vol. 134(2), pages 207-222, August.
    3. Frenk, J.B.G. & Kassay, G., 2004. "Introduction to Convex and Quasiconvex Analysis," ERIM Report Series Research in Management ERS-2004-075-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    4. Maria C. Maciel & Sandra A. Santos & Graciela N. Sottosanto, 2016. "On the Fritz John saddle point problem for differentiable multiobjective optimization," OPSEARCH, Springer;Operational Research Society of India, vol. 53(4), pages 917-933, December.
    5. Radu Boţ & Sorin-Mihai Grad & Gert Wanka, 2007. "A general approach for studying duality in multiobjective optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(3), pages 417-444, June.
    6. Fabián Flores-Bazán & William Echegaray & Fernando Flores-Bazán & Eladio Ocaña, 2017. "Primal or dual strong-duality in nonconvex optimization and a class of quasiconvex problems having zero duality gap," Journal of Global Optimization, Springer, vol. 69(4), pages 823-845, December.
    7. Frenk, J.B.G. & Schaible, S., 2004. "Fractional Programming," ERIM Report Series Research in Management ERS-2004-074-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    8. Frenk, J.B.G. & Kassay, G., 2006. "On noncooperative games, minimax theorems and equilibrium problems," Econometric Institute Research Papers EI 2006-21, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    9. Mingchao Xia & Qingying Lai & Yajiao Zhong & Canbing Li & Hsiao-Dong Chiang, 2016. "Aggregator-Based Interactive Charging Management System for Electric Vehicle Charging," Energies, MDPI, vol. 9(3), pages 1-14, March.
    10. Fernando Luque-Vásquez & J. Adolfo Minjárez-Sosa & Max E. Mitre-Báez, 2016. "A Note on König and Close Convexity in Minimax Theorems," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 65-71, July.
    11. Adan, M. & Novo, V., 2003. "Weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness," European Journal of Operational Research, Elsevier, vol. 149(3), pages 641-653, September.
    12. Frenk, J.B.G. & Schaible, S., 2004. "Fractional Programming," Econometric Institute Research Papers ERS-2004-074-LIS, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    13. C. Gutiérrez & B. Jiménez & V. Novo, 2006. "On Approximate Efficiency in Multiobjective Programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(1), pages 165-185, August.
    14. Tomás Prieto-Rumeau & José Lorenzo, 2015. "Approximation of zero-sum continuous-time Markov games under the discounted payoff criterion," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(3), pages 799-836, October.
    15. H. Boualam & A. Roubi, 2019. "Proximal bundle methods based on approximate subgradients for solving Lagrangian duals of minimax fractional programs," Journal of Global Optimization, Springer, vol. 74(2), pages 255-284, June.
    16. M. Adán & V. Novo, 2004. "Proper Efficiency in Vector Optimization on Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 121(3), pages 515-540, June.
    17. Fabián Flores-Bazán & Fernando Flores-Bazán & Cristián Vera, 2012. "A complete characterization of strong duality in nonconvex optimization with a single constraint," Journal of Global Optimization, Springer, vol. 53(2), pages 185-201, June.
    18. Frenk, J.B.G. & Kassay, G., 2005. "On Noncooperative Games and Minimax Theory," ERIM Report Series Research in Management ERS-2005-036-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    19. Frenk, J.B.G. & Kassay, G., 2006. "On Noncooperative Games, Minimax Theorems and Equilibrium Problems," ERIM Report Series Research in Management ERS-2006-022-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    20. Gutiérrez, C. & Jiménez, B. & Novo, V., 2012. "Improvement sets and vector optimization," European Journal of Operational Research, Elsevier, vol. 223(2), pages 304-311.

    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:132:y:2007:i:3:d:10.1007_s10957-007-9164-6. 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.