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Contingent epiderivatives and set-valued optimization

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  • Johannes Jahn
  • Rüdiger Rauh

Abstract

In this paper we introduce the concept of the contingent epiderivative for a set-valued map which modifies a notion introduced by Aubin [2] as upper contingent derivative. It is shown that this kind of a derivative has important properties and is one possible generalization of directional derivatives in the single-valued convex case. For optimization problems with a set-valued objective function optimality conditions based on the concept of the contingent epiderivative are proved which are necessary and sufficient under suitable assumptions. Copyright Physica-Verlag 1997

Suggested Citation

  • Johannes Jahn & Rüdiger Rauh, 1997. "Contingent epiderivatives and set-valued optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 46(2), pages 193-211, June.
    • Handle: RePEc:spr:mathme:v:46:y:1997:i:2:p:193-211
    DOI: 10.1007/BF01217690
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    Cited by:

    1. Giovanni Crespi & Ivan Ginchev & Matteo Rocca, 2006. "First-order optimality conditions in set-valued optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(1), pages 87-106, February.
    2. Luis Rodríguez-Marín & Miguel Sama, 2013. "Scalar Lagrange Multiplier Rules for Set-Valued Problems in Infinite-Dimensional Spaces," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 683-700, March.
    3. Nguyen Anh & Phan Khanh, 2013. "Higher-order optimality conditions in set-valued optimization using radial sets and radial derivatives," Journal of Global Optimization, Springer, vol. 56(2), pages 519-536, June.
    4. Abadir, Karim, 1995. "On Efficient Simulations in Dynamic Models," Discussion Papers 9521, University of Exeter, Department of Economics.
    5. Nithirat Sisarat & Rabian Wangkeeree & Tamaki Tanaka, 2020. "Sequential characterizations of approximate solutions in convex vector optimization problems with set-valued maps," Journal of Global Optimization, Springer, vol. 77(2), pages 273-287, June.
    6. Nithirat Sisarat & Rabian Wangkeeree & Gue Myung Lee, 2020. "On Set Containment Characterizations for Sets Described by Set-Valued Maps with Applications," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 824-841, March.
    7. Tran Van Su, 2023. "Optimality and duality for nonsmooth mathematical programming problems with equilibrium constraints," Journal of Global Optimization, Springer, vol. 85(3), pages 663-685, March.
    8. Luciano Fratocchi & Alberto Onetti & Alessia Pisoni & Marco Talaia, 2007. "Location of value added activities in hi-tech industries. The case of pharma-biotech firms in Italy," Economics and Quantitative Methods qf0708, Department of Economics, University of Insubria.
    9. Tran Su & Dinh Dieu Hang, 2022. "Optimality and duality in nonsmooth multiobjective fractional programming problem with constraints," 4OR, Springer, vol. 20(1), pages 105-137, March.
    10. N. L. H. Anh & P. Q. Khanh, 2013. "Variational Sets of Perturbation Maps and Applications to Sensitivity Analysis for Constrained Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 363-384, August.
    11. Crespi Giovanni P. & Ginchev Ivan & Rocca Matteo, 2004. "First order optimality conditions in set-valued optimization," Economics and Quantitative Methods qf04010, Department of Economics, University of Insubria.
    12. T. D. Chuong & J. C. Yao, 2010. "Generalized Clarke Epiderivatives of Parametric Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 146(1), pages 77-94, July.
    13. J. Baier & J. Jahn, 1999. "On Subdifferentials of Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 100(1), pages 233-240, January.
    14. Crespi Giovanni P. & Ginchev Ivan & Rocca Matteo, 2004. "First order optimality condition for constrained set-valued optimization," Economics and Quantitative Methods qf04014, Department of Economics, University of Insubria.
    15. P. Q. Khanh & N. D. Tuan, 2008. "Higher-Order Variational Sets and Higher-Order Optimality Conditions for Proper Efficiency in Set-Valued Nonsmooth Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 139(2), pages 243-261, November.
    16. S. Zhu & S. Li & K. Teo, 2014. "Second-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization," Journal of Global Optimization, Springer, vol. 58(4), pages 673-692, April.
    17. Michaël Gaydu & Michel Geoffroy & Yvesner Marcelin, 2016. "Prederivatives of convex set-valued maps and applications to set optimization problems," Journal of Global Optimization, Springer, vol. 64(1), pages 141-158, January.
    18. Elvira Hernández & Luis Rodríguez-Marín, 2011. "Weak and Strong Subgradients of Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 149(2), pages 352-365, May.
    19. Liu He & Qi-Lin Wang & Ching-Feng Wen & Xiao-Yan Zhang & Xiao-Bing Li, 2019. "A Kind of New Higher-Order Mond-Weir Type Duality for Set-Valued Optimization Problems," Mathematics, MDPI, vol. 7(4), pages 1-18, April.
    20. S. J. Li & S. K. Zhu & K. L. Teo, 2012. "New Generalized Second-Order Contingent Epiderivatives and Set-Valued Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 587-604, March.
    21. A. Taa, 2005. "ɛ-Subdifferentials of Set-valued Maps and ɛ-Weak Pareto Optimality for Multiobjective Optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 62(2), pages 187-209, November.
    22. Xiang-Kai Sun & Sheng-Jie Li, 2014. "Generalized second-order contingent epiderivatives in parametric vector optimization problems," Journal of Global Optimization, Springer, vol. 58(2), pages 351-363, February.
    23. Nguyen Thi Toan & Le Quang Thuy, 2023. "S-Derivative of the Extremum Multifunction to a Multi-objective Parametric Discrete Optimal Control Problem," Journal of Optimization Theory and Applications, Springer, vol. 196(1), pages 240-265, January.
    24. P. Q. Khanh & N. D. Tuan, 2008. "Variational Sets of Multivalued Mappings and a Unified Study of Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 139(1), pages 47-65, October.
    25. Koushik Das & Savin Treanţă & Tareq Saeed, 2022. "Mond-Weir and Wolfe Duality of Set-Valued Fractional Minimax Problems in Terms of Contingent Epi-Derivative of Second-Order," Mathematics, MDPI, vol. 10(6), pages 1-21, March.

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