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Henig proper subdifferential of set-valued maps

  • Mansi DhingraEmail author
Theoretical Article
  • 23 Downloads

Abstract

We present a notion of Henig proper subdifferential and characterize it in terms of Henig efficiency. We also present existence and some calculus rules for Henig proper subdifferentials. Using this subdifferential, we derive optimality criteria for a constrained set-valued optimization problem.

Keywords

Henig proper subdifferential Proper efficiency Set-valued map Optimality conditions 

Mathematics Subject Classification

49K30 90C46 

Notes

Acknowledgements

The author would like to express her sincere gratitude towards the anonymous referees for providing many helpful suggestions which enhanced the level of the paper. Also, the author would like to thank Prof. C.S. Lalitha, University of Delhi South Campus, New Delhi, India for providing her insight and expertise to this research work.

References

  1. 1.
    Baier, J., Jahn, J.: On subdifferentials of set-valued maps. J. Optim. Theory Appl. 100(1), 233–240 (1999)CrossRefGoogle Scholar
  2. 2.
    Chen, G.Y., Jahn, J.: Optimality conditions for set-valued optimization problems. Math. Methods Oper. Res. 48(2), 187–200 (1998)CrossRefGoogle Scholar
  3. 3.
    Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 613–630 (1968)CrossRefGoogle Scholar
  4. 4.
    Gong, X.-H.: Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior. J. Math. Anal. Appl. 307(1), 12–31 (2005)CrossRefGoogle Scholar
  5. 5.
    Ha, T.X.D.: Optimality conditions for various efficient solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium problems. Nonlinear Anal. 75(3), 1305–1323 (2012)CrossRefGoogle Scholar
  6. 6.
    Hernández, E., Rodríguez-Marín, L.: Weak and strong subgradients of set-valued maps. J. Optim. Theory Appl. 149, 352–365 (2011)CrossRefGoogle Scholar
  7. 7.
    Henig, M.I.: Proper efficiency with respect to cones. J. Optim. Theory Appl. 36(3), 387–407 (1982)CrossRefGoogle Scholar
  8. 8.
    Jahn, J.: Vector Optimization. Theory, Applications, and Extensions, 2nd edn. Springer, Berlin (2011)Google Scholar
  9. 9.
    Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Neyman, J. (ed.) Proceedings of Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 481–492. University of California Press, Berkeley (1951)Google Scholar
  10. 10.
    Lalitha, C.S., Dhingra, M.: Approximate Lagrangian duality and saddle point optimality in set optimization. RAIRO Oper. Res. 51, 819–831 (2017)CrossRefGoogle Scholar
  11. 11.
    Lalitha, C.S., Dutta, J., Govil, M.G.: Optimality criteria in set-valued optimization. J. Aust. Math. Soc. 75(2), 221–231 (2003)CrossRefGoogle Scholar
  12. 12.
    Li, S.J., Guo, X.L.: Weak subdifferential for set-valued mappings and its applications. Nonlinear Anal. 71, 5781–5789 (2009)CrossRefGoogle Scholar
  13. 13.
    Rockafellar, R.T.: Convex functions and dual extremum problems. Thesis Harvard, MA (1963)Google Scholar
  14. 14.
    Swaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization, Mathematics in Science and engineering, vol. 176. Academic Press, Orlando (1985)Google Scholar
  15. 15.
    Song, W.: Weak subdifferential of set-valued mappings. Optimization 52(3), 263–276 (2003)CrossRefGoogle Scholar
  16. 16.
    Valadier, M.: Sous-différentiabilité de fonctions convexes á valeurs dans un espace vectoriel ordonné. Math. Scand. 30, 65–74 (1972)CrossRefGoogle Scholar
  17. 17.
    Yang, X.Q.: A Hahn-Banach theorem in ordered linear spaces and its applications. Optimization 25(1), 1–9 (1992)CrossRefGoogle Scholar
  18. 18.
    Yu, G.: Generalized gradients in sense of Henig efficiency for set-valued maps. In: Proceedings of the 2009 International Joint Conference on Computational Sciences and Optimization. CSO 2009 (Volume 02), pp. 723–726 (2009)Google Scholar
  19. 19.
    Zowe, J.: Subdifferentiablity of convex functions with values in ordered vector spaces. Math. Scand. 34, 69–83 (1974)CrossRefGoogle Scholar

Copyright information

© Operational Research Society of India 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DelhiDelhiIndia

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