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Moreau–Rockafellar Theorems for Nonconvex Set-Valued Maps

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  • P. H. Sach

    (Institute of Mathematics)

Abstract

In this paper, we introduce the notion of (Benson) proper subgradient of a set-valued map and prove that, for some class of nonconvex set-valued maps, a proper subgradient of the sum of two set-valued maps can be expressed as the sum of two proper subgradients of these maps. This property is also established for weak subgradients. A result in Ref. [Lin, L.J.: J. Math. Anal. Appl. 186, 30–51 (1994)], obtained under some convexity assumption, is included as a special case of the corresponding result of this paper.

Suggested Citation

  • P. H. Sach, 2007. "Moreau–Rockafellar Theorems for Nonconvex Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 133(2), pages 213-227, May.
    • Handle: RePEc:spr:joptap:v:133:y:2007:i:2:d:10.1007_s10957-007-9173-5
    DOI: 10.1007/s10957-007-9173-5
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    References listed on IDEAS

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    1. Z. Li, 1999. "A Theorem of the Alternative and Its Application to the Optimization of Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 100(2), pages 365-375, February.
    2. P. H. Sach, 2005. "New Generalized Convexity Notion for Set-Valued Maps and Application to Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 125(1), pages 157-179, April.
    3. X. M. Yang & D. Li & S. Y. Wang, 2001. "Near-Subconvexlikeness in Vector Optimization with Set-Valued Functions," Journal of Optimization Theory and Applications, Springer, vol. 110(2), pages 413-427, August.
    4. X. M. Yang & X. Q. Yang & G. Y. Chen, 2000. "Theorems of the Alternative and Optimization with Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 107(3), pages 627-640, December.
    5. G. Y. Chen & W. D. Rong, 1998. "Characterizations of the Benson Proper Efficiency for Nonconvex Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 98(2), pages 365-384, August.
    6. T. Illés & G. Kassay, 1999. "Theorems of the Alternative and Optimality Conditions for Convexlike and General Convexlike Programming," Journal of Optimization Theory and Applications, Springer, vol. 101(2), pages 243-257, May.
    7. Z. F. Li, 1998. "Benson Proper Efficiency in the Vector Optimization of Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 98(3), pages 623-649, September.
    8. Wen Song, 1998. "A generalization of Fenchel duality in set-valued vector optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 48(2), pages 259-272, November.
    9. P. H. Sach, 2003. "Nearly Subconvexlike Set-Valued Maps and Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 119(2), pages 335-356, November.
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    Cited by:

    1. Ahmed Taa, 2019. "Subdifferential Calculus for Set-Valued Mappings and Optimality Conditions for Multiobjective Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 180(2), pages 428-441, February.

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