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Duality Theorems for Convex and Quasiconvex Set Functions

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Listed:
  • Satoshi Suzuki

    (Shimane University)

  • Daishi Kuroiwa

    (Shimane University)

Abstract

In mathematical programming, duality theorems play a central role. Especially, in convex and quasiconvex programming, Lagrange duality and surrogate duality have been studied extensively. Additionally, constraint qualifications are essential ingredients of the powerful duality theory. The best-known constraint qualifications are the interior point conditions, also known as the Slater-type constraint qualifications. A typical example of mathematical programming is a minimization problem of a real-valued function on a vector space. This types of problems have been studied widely and have been generalized in several directions. Recently, the authors investigate set functions and Fenchel duality. However, duality theorems and its constraint qualifications for mathematical programming with set functions have not been studied yet. It is expected to study set functions and duality theorems. In this paper, we study duality theorems for convex and quasiconvex set functions. We show Lagrange duality theorem for convex set functions and surrogate duality theorem for quasiconvex set functions under the Slater condition. As an application, we investigate an uncertain problem with motion uncertainty.

Suggested Citation

  • Satoshi Suzuki & Daishi Kuroiwa, 2020. "Duality Theorems for Convex and Quasiconvex Set Functions," SN Operations Research Forum, Springer, vol. 1(1), pages 1-13, March.
    • Handle: RePEc:spr:snopef:v:1:y:2020:i:1:d:10.1007_s43069-020-0005-x
    DOI: 10.1007/s43069-020-0005-x
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    References listed on IDEAS

    as
    1. Satoshi Suzuki & Daishi Kuroiwa, 2013. "Some constraint qualifications for quasiconvex vector-valued systems," Journal of Global Optimization, Springer, vol. 55(3), pages 539-548, March.
    2. Satoshi Suzuki & Daishi Kuroiwa, 2011. "On Set Containment Characterization and Constraint Qualification for Quasiconvex Programming," Journal of Optimization Theory and Applications, Springer, vol. 149(3), pages 554-563, June.
    3. Satoshi Suzuki & Daishi Kuroiwa, 2017. "Duality Theorems for Separable Convex Programming Without Qualifications," Journal of Optimization Theory and Applications, Springer, vol. 172(2), pages 669-683, February.
    4. Suzuki, Satoshi & Kuroiwa, Daishi & Lee, Gue Myung, 2013. "Surrogate duality for robust optimization," European Journal of Operational Research, Elsevier, vol. 231(2), pages 257-262.
    5. Harvey J. Greenberg & William P. Pierskalla, 1970. "Surrogate Mathematical Programming," Operations Research, INFORMS, vol. 18(5), pages 924-939, October.
    6. Satoshi Suzuki & Daishi Kuroiwa, 2012. "Necessary and Sufficient Constraint Qualification for Surrogate Duality," Journal of Optimization Theory and Applications, Springer, vol. 152(2), pages 366-377, February.
    7. Fred Glover, 1965. "A Multiphase-Dual Algorithm for the Zero-One Integer Programming Problem," Operations Research, INFORMS, vol. 13(6), pages 879-919, December.
    8. V. Jeyakumar, 2008. "Constraint Qualifications Characterizing Lagrangian Duality in Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 136(1), pages 31-41, January.
    Full references (including those not matched with items on IDEAS)

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