IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v129y2006i1d10.1007_s10957-006-9047-2.html
   My bibliography  Save this article

Fenchel-Lagrange Duality Versus Geometric Duality in Convex Optimization

Author

Listed:
  • R. I. Boţ

    (Chemnitz University of Technology)

  • S. M. Grad

    (Chemnitz University of Technology)

  • G. Wanka

    (Chemnitz University of Technology)

Abstract

We present a new duality theory to treat convex optimization problems and we prove that the geometric duality used by Scott and Jefferson in different papers during the last quarter of century is a special case of it. Moreover, weaker sufficient conditions to achieve strong duality are considered and optimality conditions are derived. Next, we apply our approach to some problems considered by Scott and Jefferson, determining their duals. We give weaker sufficient conditions to achieve strong duality and the corresponding optimality conditions. Finally, posynomial geometric programming is viewed also as a particular case of the duality approach that we present.

Suggested Citation

  • R. I. Boţ & S. M. Grad & G. Wanka, 2006. "Fenchel-Lagrange Duality Versus Geometric Duality in Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 129(1), pages 33-54, April.
    • Handle: RePEc:spr:joptap:v:129:y:2006:i:1:d:10.1007_s10957-006-9047-2
    DOI: 10.1007/s10957-006-9047-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-006-9047-2
    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/s10957-006-9047-2?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. Elmor L. Peterson, 1976. "Fenchel's Duality Thereom in Generalized Geometric Programming," Discussion Papers 252, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    2. Elmor L. Peterson, 1976. "Optimality Conditions in Generalized Geometric Programming," Discussion Papers 221, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    3. C.H. Scott & T.R. Jefferson, 2003. "On Duality for a Class of Quasiconcave Multiplicative Programs," Journal of Optimization Theory and Applications, Springer, vol. 117(3), pages 575-583, June.
    4. R. I. Boţ & G. Kassay & G. Wanka, 2005. "Strong Duality for Generalized Convex Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 127(1), pages 45-70, October.
    5. Gert Wanka & Radu-Ioan Boţ, 2001. "Multiobjective duality for convex-linear problems II," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 53(3), pages 419-433, July.
    6. T.R. Jefferson & C.H. Scott, 2001. "Quality Tolerancing and Conjugate Duality," Annals of Operations Research, Springer, vol. 105(1), pages 185-200, July.
    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. M. D. Fajardo & J. Vidal, 2018. "Necessary and Sufficient Conditions for Strong Fenchel–Lagrange Duality via a Coupling Conjugation Scheme," Journal of Optimization Theory and Applications, Springer, vol. 176(1), pages 57-73, January.
    2. Milan Hladík, 2011. "Optimal value bounds in nonlinear programming with interval data," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(1), pages 93-106, July.
    3. R. I. Boţ & S. M. Grad & G. Wanka, 2007. "New Constraint Qualification and Conjugate Duality for Composed Convex Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 135(2), pages 241-255, November.

    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. C.H. Scott & T.R. Jefferson, 2003. "On Duality for a Class of Quasiconcave Multiplicative Programs," Journal of Optimization Theory and Applications, Springer, vol. 117(3), pages 575-583, June.
    2. Chiu, Nan-Chieh & Fang, Shu-Cherng & Lavery, John E. & Lin, Jen-Yen & Wang, Yong, 2008. "Approximating term structure of interest rates using cubic L1 splines," European Journal of Operational Research, Elsevier, vol. 184(3), pages 990-1004, February.
    3. Kenneth Lange & Eric C. Chi & Hua Zhou, 2014. "Rejoinder," International Statistical Review, International Statistical Institute, vol. 82(1), pages 81-89, April.
    4. C. Scott & T. Jefferson, 2007. "On duality for square root convex programs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(1), pages 75-84, February.
    5. C. E. Gounaris & C. A. Floudas, 2008. "Convexity of Products of Univariate Functions and Convexification Transformations for Geometric Programming," Journal of Optimization Theory and Applications, Springer, vol. 138(3), pages 407-427, September.
    6. Belleh Fontem, 2023. "Robust Chance-Constrained Geometric Programming with Application to Demand Risk Mitigation," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 765-797, May.
    7. Hua Zhou & Kenneth Lange, 2015. "Path following in the exact penalty method of convex programming," Computational Optimization and Applications, Springer, vol. 61(3), pages 609-634, July.
    8. M. D. Fajardo & J. Vidal, 2018. "Necessary and Sufficient Conditions for Strong Fenchel–Lagrange Duality via a Coupling Conjugation Scheme," Journal of Optimization Theory and Applications, Springer, vol. 176(1), pages 57-73, January.
    9. R. I. Boţ & S. M. Grad & G. Wanka, 2007. "Fenchel’s Duality Theorem for Nearly Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 132(3), pages 509-515, March.
    10. Kasina, Saamrat & Hobbs, Benjamin F., 2020. "The value of cooperation in interregional transmission planning: A noncooperative equilibrium model approach," European Journal of Operational Research, Elsevier, vol. 285(2), pages 740-752.
    11. S. K. Mishra & Vinay Singh & Vivek Laha, 2016. "On duality for mathematical programs with vanishing constraints," Annals of Operations Research, Springer, vol. 243(1), pages 249-272, August.
    12. Yalçın Küçük & İlknur Atasever & Mahide Küçük, 2012. "Weak Fenchel and weak Fenchel-Lagrange conjugate duality for nonconvex scalar optimization problems," Journal of Global Optimization, Springer, vol. 54(4), pages 813-830, December.
    13. R. I. Boţ & S. M. Grad & G. Wanka, 2007. "New Constraint Qualification and Conjugate Duality for Composed Convex Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 135(2), pages 241-255, November.
    14. Qingjie Hu & Jiguang Wang & Yu Chen, 2020. "New dualities for mathematical programs with vanishing constraints," Annals of Operations Research, Springer, vol. 287(1), pages 233-255, April.
    15. C. R. Chen & S. J. Li, 2009. "Different Conjugate Dual Problems in Vector Optimization and Their Relations," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 443-461, March.
    16. Najafi, Arsalan & Homaee, Omid & Jasiński, Michał & Tsaousoglou, Georgios & Leonowicz, Zbigniew, 2023. "Integrating hydrogen technology into active distribution networks: The case of private hydrogen refueling stations," Energy, Elsevier, vol. 278(PB).
    17. Li, S.J. & Chen, C.R. & Wu, S.Y., 2009. "Conjugate dual problems in constrained set-valued optimization and applications," European Journal of Operational Research, Elsevier, vol. 196(1), pages 21-32, July.

    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:129:y:2006:i:1:d:10.1007_s10957-006-9047-2. 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.