IDEAS home Printed from https://ideas.repec.org/a/spr/snopef/v1y2020i1d10.1007_s43069-020-0005-x.html
   My bibliography  Save this article

Duality Theorems for Convex and Quasiconvex Set Functions

Author

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s43069-020-0005-x
    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/s43069-020-0005-x?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. 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.
    2. 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.
    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. Fred Glover, 1965. "A Multiphase-Dual Algorithm for the Zero-One Integer Programming Problem," Operations Research, INFORMS, vol. 13(6), pages 879-919, December.
    6. 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.
    7. Harvey J. Greenberg & William P. Pierskalla, 1970. "Surrogate Mathematical Programming," Operations Research, INFORMS, vol. 18(5), pages 924-939, October.
    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)

    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. Satoshi Suzuki, 2019. "Optimality Conditions and Constraint Qualifications for Quasiconvex Programming," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 963-976, December.
    2. 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.
    3. Satoshi Suzuki, 2021. "Karush–Kuhn–Tucker type optimality condition for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential," Journal of Global Optimization, Springer, vol. 79(1), pages 191-202, January.
    4. 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.
    5. 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.
    6. Marco Antonio Boschetti & Vittorio Maniezzo, 2022. "Matheuristics: using mathematics for heuristic design," 4OR, Springer, vol. 20(2), pages 173-208, June.
    7. Alidaee, Bahram, 2014. "Zero duality gap in surrogate constraint optimization: A concise review of models," European Journal of Operational Research, Elsevier, vol. 232(2), pages 241-248.
    8. Yoon, Yourim & Kim, Yong-Hyuk & Moon, Byung-Ro, 2012. "A theoretical and empirical investigation on the Lagrangian capacities of the 0-1 multidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 218(2), pages 366-376.
    9. 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.
    10. Satoshi Suzuki & Daishi Kuroiwa, 2015. "Characterizations of the solution set for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential," Journal of Global Optimization, Springer, vol. 62(3), pages 431-441, July.
    11. Ablanedo-Rosas, José H. & Rego, César, 2010. "Surrogate constraint normalization for the set covering problem," European Journal of Operational Research, Elsevier, vol. 205(3), pages 540-551, September.
    12. Edirisinghe, Chanaka & Jeong, Jaehwan, 2019. "Indefinite multi-constrained separable quadratic optimization: Large-scale efficient solution," European Journal of Operational Research, Elsevier, vol. 278(1), pages 49-63.
    13. Renaud Chicoisne, 2023. "Computational aspects of column generation for nonlinear and conic optimization: classical and linearized schemes," Computational Optimization and Applications, Springer, vol. 84(3), pages 789-831, April.
    14. Yuji Nakagawa & Ross J. W. James & César Rego & Chanaka Edirisinghe, 2014. "Entropy-Based Optimization of Nonlinear Separable Discrete Decision Models," Management Science, INFORMS, vol. 60(3), pages 695-707, March.
    15. T. D. Chuong & V. Jeyakumar & G. Li, 2019. "A new bounded degree hierarchy with SOCP relaxations for global polynomial optimization and conic convex semi-algebraic programs," Journal of Global Optimization, Springer, vol. 75(4), pages 885-919, December.
    16. Patriksson, Michael, 2008. "A survey on the continuous nonlinear resource allocation problem," European Journal of Operational Research, Elsevier, vol. 185(1), pages 1-46, February.
    17. Saïd Hanafi & Christophe Wilbaut, 2011. "Improved convergent heuristics for the 0-1 multidimensional knapsack problem," Annals of Operations Research, Springer, vol. 183(1), pages 125-142, March.
    18. Jae Hyoung Lee & Gue Myung Lee, 2018. "On optimality conditions and duality theorems for robust semi-infinite multiobjective optimization problems," Annals of Operations Research, Springer, vol. 269(1), pages 419-438, October.
    19. Johri, Pravin K., 1996. "Implied constraints and an alternate unified development of nonlinear programming theory," European Journal of Operational Research, Elsevier, vol. 88(3), pages 537-549, February.
    20. Thomas L. Magnanti, 2021. "Optimization: From Its Inception," Management Science, INFORMS, vol. 67(9), pages 5349-5363, September.

    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:snopef:v:1:y:2020:i:1:d:10.1007_s43069-020-0005-x. 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.