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Lagrangian Duality and Cone Convexlike Functions

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  • J. B. G. Frenk

    (Erasmus University Rotterdam)

  • G. Kassay

    (Babes-Bolyai University)

Abstract

In this paper, we consider first the most important classes of cone convexlike vector-valued functions and give a dual characterization for some of these classes. It turns out that these characterizations are strongly related to the closely convexlike and Ky Fan convex bifunctions occurring within minimax problems. Applying the Lagrangian perturbation approach, we show that some of these classes of cone convexlike vector-valued functions show up naturally in verifying strong Lagrangian duality for finite-dimensional optimization problems. This is achieved by extending classical convexity results for biconjugate functions to the class of so-called almost convex functions. In particular, for a general class of finite-dimensional optimization problems, strong Lagrangian duality holds if some vector-valued function related to this optimization problem is closely K-convexlike and satisfies some additional regularity assumptions. For K a full-dimensional convex cone, it turns out that the conditions for strong Lagrangian duality simplify. Finally, we compare the results obtained by the Lagrangian perturbation approach worked out in this paper with the results achieved by the so-called image space approach initiated by Giannessi.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:134:y:2007:i:2:d:10.1007_s10957-007-9221-1
    DOI: 10.1007/s10957-007-9221-1
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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. 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.
    3. J.F.B. Frenk & G. Kassay, 2002. "Minimax Results and Finite-Dimensional Separation," Journal of Optimization Theory and Applications, Springer, vol. 113(2), pages 409-421, May.
    4. 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.
    5. G. Mastroeni & T. Rapcsák, 2000. "On Convex Generalized Systems," Journal of Optimization Theory and Applications, Springer, vol. 104(3), pages 605-627, March.
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    Cited by:

    1. G. Mastroeni, 2010. "Some applications of the image space analysis to the duality theory for constrained extremum problems," Journal of Global Optimization, Springer, vol. 46(4), pages 603-614, April.
    2. Jean-Paul Penot, 2010. "Are dualities appropriate for duality theories in optimization?," Journal of Global Optimization, Springer, vol. 47(3), pages 503-525, July.
    3. G. Y. Li, 2011. "A Note on Nonconvex Minimax Theorem with Separable Homogeneous Polynomials," Journal of Optimization Theory and Applications, Springer, vol. 150(1), pages 194-203, July.

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