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Higher-Order Efficiency Conditions for Continuously Directional Differentiable Vector Equilibrium Problem with Constraints

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Abstract

In this paper, we give optimality conditions of higher-order for a continuously directional differentiable vector equilibrium problem with constraints. First, we obtain some important characterizations and existence results for a m-times continuously directional differentiable function(m is a positive integer number). Second, as an application, we provide some higher-order Karush–Kuhn–Tucker necessary optimality conditions for efficiency. Finally, some higher-order sufficient optimality conditions, which are very close to the higher-order Karush–Kuhn–Tucker necessary optimality conditions, are presented as well. Some illustrative examples are also given.

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References

  1. Constantin, E.: Higher-order sufficient conditions for problems with G\(\widehat{a}\)teaux differentiable data. Revue Roumaini de Mathématique Pures and Appliquées, Tome LXIV, vol. 1 (2019)

  2. Constantin, E.: Optimization and Flow Invariance via High Order Tangent Cones. Ph.D Dissertation. Ohio University (2005)

  3. Constantin, E.: Higher order necessary and sufficient conditions for optimality. Panam. Math. J. 14, 1–25 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Constantin, E.: Second-order necessary conditions based on second-order tangent cones. Math. Sci. Res. J. 10(2), 42–56 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Constantin, E.: Higher order necessary conditions in smooth constrained optimization. Commun. Math. 479, 41–49 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Constantin, E.: Second-order necessary conditions for set constrained nonsmooth optimization problems via second-order projective tangent cones. Lib. Math. 36(1), 1–24 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Constantin, E.: Necessary conditions for weak efficiency for nonsmooth degenerate multiobjective optimization problems. J. Glob. Optim. 75(1), 111–129 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dinh, N., Goberna, M.A., Long, D.H., Volle, M.: Duality for constrained robust sum optimization problems. Math. Program. (2020). https://doi.org/10.1007/s10107-020-01494-1

  9. Ginchev, I., Guerraggio, A., Rocca, M.: Higher order properly efficient points in vector optimization. In: Generalized Convexity and Related Topics, Lecture Notes in Econum. and Math. Systems. vol. 583, pp. 227–245. Springer, Berlin (2007)

  10. Ginchev, I.: Higher-order optimality conditions in nonsmooth optimization. Optimization 51, 47–52 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ginchev, I., Guerraggio, A., Rocca, M.: Geoffrion type characterization of higher-order properly efficient points in vector optimization. J. Math. Anal. Appl. 328(2), 780–788 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gong, X.H.: Optimality conditions for vector equilibrium problems. J. Math. Anal. Appl. 342, 1455–1466 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gong, X.H.: Scalarization and optimality conditions for vector equilibrium problems. Nonlinear Anal. 73, 3598–3612 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Guerraggio, A., Luc, D.T.: Properly maximal points in product spaces. Math. Oper. Res. 31, 305–315 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gutiérrez, C., Jiménez, B., Novo, V.: On second-order Fritz John type optimality conditions in nonsmooth multiobjective programming. Math. Program. 123(B), 199–223 (2010)

  16. Ivanov, V.I.: Higher-order optimality conditions for inequality constrained problems. Appl. Anal. 92, 2600–2617 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ivanov, V.I.: Second-order optimality conditions for vector problems with continuously Fréchet differentiable data and second-order constraint qualifications. J. Optim. Theory Appl. 166, 777–790 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ivanov, V.I.: Higher-order optimality conditions with an arbitrary nondifferentiable function. Optimization 65(11), 1909–1927 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jahn, J.: Theory, Applications and Extensions Second Edition Vector Optimization. Springer, Berlin (2011)

    MATH  Google Scholar 

  20. Jiménez, B., Novo, V.: First- and second-order sufficient conditions for strict minimality in nonsmooth vector optimization. J. Math. Anal. Appl. 284, 496–510 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jiménez, B., Novo, V.: First order optimality conditions in vector optimization involving stable functions. Optimization 57(3), 449–471 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Khan, A., Tammer, C., Zalinescu, C.: Set-Valued Optimization. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  23. Khanh, P.Q., Tung, N.M.: Optimality conditions and duality for nonsmooth vector equilibrium problems with constraints. Optimization 64, 1547–1575 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lee, H., Pavel, N.: Higher order optimality conditions and its applications. Panam. Math. J. 14, 11–24 (2004)

    MathSciNet  MATH  Google Scholar 

  25. Luc, D.T., Volle, M.: Duality for optimization problems with infinite sums. SIAM. J. Optim. 29(3), 1819–1843 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  26. Luu, D.V.: Higher-order necessary and sufficient conditions for strict local Pareto minima in terms of Studniarski’s derivatives. Optimization 57, 593–605 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Luu, D.V.: Higher-order optimality conditions in nonsmooth cone-constrained multiobjective programming. Nonlinear Funct. Anal. Appl. 15, 381–393 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Luu, D.V.: Higher-order efficiency conditions via higher-order tangent cones. Numer. Funct. Anal. Optim. 35, 68–84 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Motreanu, D., Pavel, N.H.: Tangency, Flow Invariance for Differential Equations and Optimization Problems, Monographs and Textbooks in Pure and Appl. Mathematics, vol. 219. Marcel Dekker, New York (1999)

  30. Pavel, N.H., Ursescu, C.: Flow-invariant sets for autonomous second order differential equations and applications in mechanics. Nonlinear Anal. 6, 35–77 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  31. Penot, J.P.: Second-order conditions for optimization problems with constraints. SIAM J. Control Optim. 37, 303–318 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  32. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

  33. Studniarski, M.: Necessary and sufficient conditions for isolated local minima of nonsmooth functions. SIAM J. Control Optim. 24, 1044–1049 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  34. Su, T.V., Hang, D.D.: Second-order necessary and sufficient optimality conditions for constrained vector equilibrium problem with applications. Bull. Iran. Math. Soc. (2020). https://doi.org/10.1007/s41980-020-00445-y

  35. Su, T.V., Luu, D,V.: Higher-order efficiency conditions for constrained vector equilibrium problems. Optimization (2021). https://doi.org/10.1080/02331934.2021.1873987

  36. Su, T.V., Luu, D.V.: Higher-order Karush–Kuhn–Tucker optimality conditions for Borwein properly efficient solutions of multiobjective semi-infinite programming. Optimization (2020). https://doi.org/10.1080/02331934.2020.1836633

  37. Su, T.V.: New second-order optimality conditions for vector equilibrium problems with constraints in terms of contingent derivatives. Bull. Braz. Math. Soc. New Ser. 51(2), 371–395 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  38. Su, T.V., Hang, D.D.: On optimality conditions for efficient solutions in constrained vector equilibrium problems in terms of Studniarski derivatives. J. Nonlinear Funct. Anal. 27, 1–19 (2020)

    Google Scholar 

  39. Su, T.V., Hien, N.D.: Studniarski’s derivatives and efficiency conditions for constrained vector equilibrium problems with applications. Optimization 70(1), 121–148 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  40. Taa, A.: Second order conditions for nonsmooth multiobjective optimization problems with inclusion constrains. J. Glob. Optim. 50, 271–291 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. Tung, N.M.: New higher-order strong Karush–Kuhn–Tucker conditions for proper solutions in nonsmooth optimization. J. Optim. Theory Appl. 185, 448–475 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ursescu, C.: Tangent sets’ calculus and necessary conditions for extremality. SIAM J. Control Optim. 20, 563–574 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author is indebted to the two anonymous referees for very useful comments and suggestions on the preceding versions of this paper. The author research is supported by Thai Nguyen University of Information and Communication Technology [grant number DH2021-TN07-01].

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  1. Department of Mathematics, Quang Nam University, 102 Hung Vuong, Tamky, Vietnam

    Tran Van Su

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  1. Tran Van Su
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Communicated by Behzad Djafari-Rouhani.

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Van Su, T. Higher-Order Efficiency Conditions for Continuously Directional Differentiable Vector Equilibrium Problem with Constraints. Bull. Iran. Math. Soc. 48, 1805–1828 (2022). https://doi.org/10.1007/s41980-021-00621-8

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  • DOI: https://doi.org/10.1007/s41980-021-00621-8

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