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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
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    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.
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    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.

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