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Second-order weakly composed adjacent-generalized contingent epiderivatives and applications to composite set-valued optimization problems

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Abstract

In the paper, we introduce the second-order weakly composed adjacent-generalized contingent epiderivative for set-valued maps. Then we gain a few crucial properties of the epiderivative. Moreover, we obtain the sum and chain rules of the epiderivative. Finally, by virtue of the epiderivative, we establish the necessary optimality conditions and sufficient optimality conditions for Benson proper efficient solutions of unconstrained composed set-valued optimization problems. The main results of this paper are illustrated by many examples.

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References

  1. Anh, N.L.H.: Higher-order optimality conditions in set-valued optimization using Studniarski derivatives and applications to duality. Positivity 18, 449–473 (2014)

    Article  MathSciNet  Google Scholar 

  2. Aubin, J.P.: Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions. In: Nachbin, L. (ed.) Mathematical Analysis and Applications, Part A, pp. 160–229. Academic Press, New York (1981)

    Google Scholar 

  3. Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)

    MATH  Google Scholar 

  4. Aubin, J.P., Frankowska, H.: Set-valued Analysis. Birkhauser, Boston (1990)

    MATH  Google Scholar 

  5. Benson, H.P.: An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Anal. Appl. 71, 232–241 (1979)

    Article  MathSciNet  Google Scholar 

  6. Bigi, G., Castellani, M.: K-epiderivatives for set-valued functions and optimization. Math. Methods Oper. Res. 55, 401–412 (2002)

    Article  MathSciNet  Google Scholar 

  7. Bigi, G.: On sufficient second order optimality conditions in multiobjective optimization. Math. Methods Oper. Res. 63, 77–85 (2006)

    Article  MathSciNet  Google Scholar 

  8. Borwein, J.M., Zhuang, D.: Super efficiency in vector optimization. Trans. Am. Math. Soc. 338, 105–122 (1993)

    Article  MathSciNet  Google Scholar 

  9. Boţ, R.I., Grad, S.M., Wanka, G.: New constraint qualification and conjugate duality for composed convex optimization Problems. J. Optim. Theory Appl. 135, 241–255 (2007)

    Article  MathSciNet  Google Scholar 

  10. Boţ, R.I., Hodrea, I.B., Wanka, G.: Optimality conditions for weak efficiency to vector optimization problems with composed convex functions. Cent. Eur. J. Math. 6, 453–468 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Chen, G.Y., Rong, W.D.: Characterizations of the Benson proper efficiency for nonconvex vector optimization. J. Optim. Theory Appl. 98, 365–384 (1998)

    Article  MathSciNet  Google Scholar 

  13. Chen, C.R., Li, S.J., Teo, K.L.: Higher order weak epiderivatives and applications to duality and optimality conditions. Comput. Math. Appl. 57, 1389–1399 (2009)

    Article  MathSciNet  Google Scholar 

  14. Corley, H.W.: Optimality condition for maximizations of set-valued functions. J. Optim. Theory Appl. 58, 1–10 (1988)

    Article  MathSciNet  Google Scholar 

  15. Fang, D.H., Wang, X.Y.: Stable and total Fenchel duality for composed convex optimization problems. Acta Math. Appl. Sin. Engl. Ser. 34, 813–827 (2018)

    Article  MathSciNet  Google Scholar 

  16. Flores-Bazàn, F.: Optimality conditions in nonconvex set-valued optimization. Math. Methods Oper. Res. 53, 403–417 (2001)

    Article  MathSciNet  Google Scholar 

  17. He, L., Wang, Q.L., Wen, C.F., et al.: A kind of new higher-order Mond–Weir type duality for set-valued optimization problems. Mathematics 7, 1–18 (2019)

    Google Scholar 

  18. Isac, G., Khan, A.A.: Second-order optimality conditions in set-valued optimization by a new tangential derivative. Acta Math. Vietnam. 34, 81–90 (2009)

    MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

  20. Jahn, J., Khan, A.A., Zeilinger, P.: Second-order optimality conditions in set optimization. J. Optim. Theory Appl. 125, 331–347 (2005)

    Article  MathSciNet  Google Scholar 

  21. Jiménez, B., Novo, V.: Second-order necessary conditions in set constrained differentiable vector optimization. Math. Methods Oper. Res. 58, 299–317 (2003)

    Article  MathSciNet  Google Scholar 

  22. Jiménez, B., Novo, V.: Optimality conditions in differentiable vector optimization via second-order tangent sets. Appl. Math. Optim. 49, 123–144 (2004)

    Article  MathSciNet  Google Scholar 

  23. Lalitha, C.S., Arora, R.: Weak Clarke epiderivative in set-valued optimization. J. Math. Anal. Appl. 342, 704–714 (2008)

    Article  MathSciNet  Google Scholar 

  24. Li, S.J., Chen, C.R.: Higher order optimality conditions for Henig efficient solutions in set-valued optimization. J. Math. Anal. Appl. 323, 1184–1200 (2006)

    Article  MathSciNet  Google Scholar 

  25. Li, S.J., Teo, K.L., Yang, X.Q.: Higher-order optimality conditions for set-valued optimization. J. Optim. Theory Appl. 137, 533–553 (2008)

    Article  MathSciNet  Google Scholar 

  26. Li, S.J., Zhu, S.K., Teo, K.L.: New generalized second-order contingent epiderivatives and set-valued optimization problems. J. Optim. Theory Appl. 152, 587–604 (2012)

    Article  MathSciNet  Google Scholar 

  27. Long, X.J., Sun, X.K., Peng, Z.Y.: Approximate optimality conditions for composite convex optimization problems. J. Oper. Res. Soc. China 5, 469–485 (2017)

    Article  MathSciNet  Google Scholar 

  28. Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989)

    Book  Google Scholar 

  29. Peng, Z.H., Xu, Y.H.: New second-order tangent epiderivatives and applications to set-valued optimization. J. Optim. Theory Appl. 172, 128–140 (2017)

    Article  MathSciNet  Google Scholar 

  30. Peng, Z.H., Wan, Z.P.: Second-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization subject to mixed constraints. Results Math. 73, 1–20 (2018)

    Article  MathSciNet  Google Scholar 

  31. Peng, Z.H., Wan, Z.P., Guo, Y.J.: New higher-order weak lower inner epiderivatives and application to Karush-Kuhn–Tucker necessary optimality conditions in set-valued optimization. Jpn. J. Ind. Appl. Math. 37, 851–866 (2020)

    Article  MathSciNet  Google Scholar 

  32. Penot, J.P.: Second-order conditions for optimization problems with constraints. SIAM J. Control Optim. 37, 303–318 (1998)

    Article  MathSciNet  Google Scholar 

  33. Wang, Q.L., Li, S.J.: Generalized higher-order optimality conditions for set-valued optimization under Henig efficiency. Numer. Funct. Anal. Optim. 30, 849–869 (2009)

    Article  MathSciNet  Google Scholar 

  34. Wang, Q.L., Yu, G.L.: Higher-order weakly generalized epiderivatives and applications to optimality conditions. J. Appl. Math. 2012, 2607–2645 (2012)

    MathSciNet  MATH  Google Scholar 

  35. Wang, Q.L., Li, X.B., Yu, G.L.: Second-order weak composed epiderivatives and applications to optimality conditions. J. Ind. Manag. Optim. 9, 455–470 (2013)

    MathSciNet  MATH  Google Scholar 

  36. Wang, Q.L., Zhang, X.Y., Li, X.B., et al.: New second-order radial epiderivatives and nonconvex set-valued optimization problems. Jpn. J. Ind. Appl. Math. 36, 907–935 (2019)

    Article  MathSciNet  Google Scholar 

  37. Wang, Q.L., Zhang, X.Y.: Second-order composed radial derivatives of the Benson proper perturbation map for parametric multi-objective optimization problems. Asia Pac. J. Oper. Res. 37(1–18), 2040011 (2020)

    Article  MathSciNet  Google Scholar 

  38. Zhang, X.Y., Wang, Q.L.: New second-order radial epiderivatives and applications to optimality conditions. RAIRO-Oper. Res. 54, 949–959 (2019)

    Article  MathSciNet  Google Scholar 

  39. Zheng, X.Y.: Proper efficiency in locally convex topological vector spaces. J. Optim. Theory Appl. 94, 469–486 (1997)

    Article  MathSciNet  Google Scholar 

  40. Zhu, S.K., Li, S.J., Teo, K.L.: Second-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization. J. Glob. Optim. 58, 673–692 (2014)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank anonymous referees for their valuable comments and suggestions, which helped to improve the paper. This research was partially supported by the National Natural Science Foundation of China (No.11971078), Team Building Project for Graduate Tutors in Chongqing (JDDSTD201802), Postgraduate Research and Innovation Project of Chongqing (CYS21372) and the Group Building Project for Scientific Innovation for Universities in Chongqing (CXQT21021).

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  1. College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 400074, China

    Tian Tang, Qilin Wang, Xiaoyan Zhang & Yuwen Zhai

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  1. Tian Tang
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  2. Qilin Wang
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  3. Xiaoyan Zhang
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  4. Yuwen Zhai
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Correspondence to Qilin Wang.

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Tang, T., Wang, Q., Zhang, X. et al. Second-order weakly composed adjacent-generalized contingent epiderivatives and applications to composite set-valued optimization problems. Japan J. Indust. Appl. Math. 39, 319–350 (2022). https://doi.org/10.1007/s13160-021-00491-x

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  • DOI: https://doi.org/10.1007/s13160-021-00491-x

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