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Weak Subdifferentials for Set-Valued Mappings

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Abstract

The purpose of this paper is to study the weak subdifferential for set-valued mappings, which was introduced by Chen and Jahn (Math. Methods Oper. Res., 48:187–200, 1998). Two existence theorems of weak subgradients for set-valued mappings are obtained. Moreover, some properties of the weak subdifferential for set-valued mappings are derived. Our results improve the corresponding ones in the literature. Some examples are given to illustrate our results.

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References

  1. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1979)

    Google Scholar 

  2. Chen, G.Y., Jahn, J.: Optimality conditions for set-valued optimization problems. Math. Methods Oper. Res. 48, 187–200 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  4. Baier, J., Jahn, J.: On subdifferentials of set-valued maps. J. Optim. Theory Appl. 100, 233–240 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Borwein, J.M.: A Lagrange multiplier theorem and a sandwich theorem for convex relations. Math. Scand. 48, 189–204 (1981)

    MATH  MathSciNet  Google Scholar 

  6. Jahn, J.: Vector Optimization: Theory, Applications, and Extensions. Springer, Berlin (2004)

    Book  Google Scholar 

  7. Song, W.: Weak subdifferential of set-valued mappings. Optimization 52, 263–276 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Tanino, T.: Conjugate duality in vector optimization. J. Math. Anal. Appl. 167, 84–97 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen, G.Y., Craven, B.D.: A vector variational inequality and optimization over an efficient set. Math. Methods Oper. Res. 34, 1–12 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  10. Yang, X.Q.: A Hahn–Banach theorem in ordered linear spaces and its applications. Optimization 25, 1–9 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  11. Peng, W.J., Lee, H.W.J., Rong, W.D., Yang, X.M.: Hahn-Banach theorems and subgradients of set-valued maps. Math. Methods Oper. Res. 61, 281–297 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Li, S.J., Guo, X.L.: Weak subdifferential for set-valued mappings and it applications. Nonlinear Anal. 71, 5781–5789 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Zalinescu, C.: Hahn-Banach extension theorems for multifunctions revisited. Math. Methods Oper. Res. 68, 493–508 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hernandez, E., Rodriguez-Marin, L.: Weak and strong subgradients of set-valued maps. J. Optim. Theory Appl. 149, 352–365 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Bouligand, G.: Sur l’existence des demi-tangentes á une courbe de Jordan. Fundam. Math. 15, 215 (1930)

    MATH  Google Scholar 

  16. Taa, A.: Necessary and sufficient conditions for multiobjective optimization problems. Optimization 36, 97–104 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  17. Amahroq, T., Thibault, L.: On proto-differentiability and strict proto-differentiability of multifunctions of feasible points in perturbed optimization problems. Numer. Funct. Anal. Optim. 16, 1293–1307 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  18. Breckner, W.W., Kassay, G.: A systematization of convexity concepts for sets and functions. J. Convex Anal. 4, 109–127 (1997)

    MATH  MathSciNet  Google Scholar 

  19. Taa, A.: Set-valued derivatives of multifunctions and optimality conditions. Numer. Funct. Anal. Optim. 19, 121–149 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the editor and the anonymous referees for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (11001287, 11171363 and 11201509), the Natural Science Foundation Project of Chongqing (CSTC 2010BB9254 and CSTC 2009BB8240), the Education Committee Project Research Foundation of Chongqing (KJ100711), the Special Fund of Chongqing Key Laboratory (CSTC 2011KLORSE01) and the project of the third batch support program for excellent talents of Chongqing City High Colleges.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, P.R. China

    X. J. Long

  2. Department of Mathematics, Chongqing Normal University, Chongqing, 400047, P.R. China

    J. W. Peng

  3. College of Science, Chongqing JiaoTong University, Chongqing, 400074, P.R. China

    X. B. Li

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  1. X. J. Long
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  2. J. W. Peng
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  3. X. B. Li
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Correspondence to X. J. Long.

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Communicated by Jafar Zafarani.

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Long, X.J., Peng, J.W. & Li, X.B. Weak Subdifferentials for Set-Valued Mappings. J Optim Theory Appl 162, 1–12 (2014). https://doi.org/10.1007/s10957-013-0469-3

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