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New Constraint Qualification and Conjugate Duality for Composed Convex Optimization Problems

Author

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  • 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 constraint qualification which guarantees strong duality between a cone-constrained convex optimization problem and its Fenchel-Lagrange dual. This result is applied to a convex optimization problem having, for a given nonempty convex cone K, as objective function a K-convex function postcomposed with a K-increasing convex function. For this so-called composed convex optimization problem, we present a strong duality assertion, too, under weaker conditions than the ones considered so far. As an application, we rediscover the formula of the conjugate of a postcomposition with a K-increasing convex function as valid under weaker conditions than usually used in the literature.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:135:y:2007:i:2:d:10.1007_s10957-007-9247-4
    DOI: 10.1007/s10957-007-9247-4
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    References listed on IDEAS

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    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. M. Volle, 2002. "Duality Principles for Optimization Problems Dealing with the Difference of Vector-Valued Convex Mappings," Journal of Optimization Theory and Applications, Springer, vol. 114(1), pages 223-241, July.
    3. 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.
    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.
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    Cited by:

    1. D. H. Fang & Y. Zhang, 2018. "Extended Farkas’s Lemmas and Strong Dualities for Conic Programming Involving Composite Functions," Journal of Optimization Theory and Applications, Springer, vol. 176(2), pages 351-376, February.
    2. Nguyen Dinh & Dang Hai Long, 2022. "A Perturbation Approach to Vector Optimization Problems: Lagrange and Fenchel–Lagrange Duality," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 713-748, August.
    3. Bot, Radu Ioan & Lorenz, Nicole, 2011. "Optimization problems in statistical learning: Duality and optimality conditions," European Journal of Operational Research, Elsevier, vol. 213(2), pages 395-404, September.
    4. N. Dinh & V. Jeyakumar, 2014. "Farkas’ lemma: three decades of generalizations for mathematical optimization," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 1-22, April.

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