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Second-order necessary efficiency conditions for nonsmooth vector equilibrium problems

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Abstract

This paper presents primal and dual second-order Fritz John necessary conditions for weak efficiency of nonsmooth vector equilibrium problems involving inequality, equality and set constraints in terms of the Páles–Zeidan second-order directional derivatives. Dual second-order Karush–Kuhn–Tucker necessary conditions for weak efficiency are established under suitable second-order constraint qualifications.

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References

  1. Aghezzaf, B., Hachimi, M.: Second-order optimality conditions in multiobjective optimization problems. J. Optim. Theory Appl. 102(1), 37–50 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ben-Tal, A.: Second-order and related extremality conditions in nonlinear programming. J. Optim. Theory Appl. 31(2), 143–165 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blum, E., Oettli, W.: From optimization and variational inequalit ies to equilibrium problems. Math. Stud. 63, 127–149 (1994)

    Google Scholar 

  4. Constantin, E.: Second-order necessary conditions in locally Lipschitz optimization with inequality constraints. Optim. Lett. 9, 245–261 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley Interscience, New York (1983)

    MATH  Google Scholar 

  7. Daniele, P.: Lagrange multipliers and infinite-dimensional equilibrium problems. J. Glob. Optim. 40, 65–70 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Daniele, P.: Dynamic Networks and Evolutionary Variational Inequalities. Edward Elgar Publishing, Cheltenham (2006)

    MATH  Google Scholar 

  9. Giannessi, F., Mastroeni, G., Pellegrini, L.: On the theory of vector optimization and variational inequalities, image space analysis and separation. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, pp. 153–215. Kluwer, Dordrecht (2000)

    Chapter  Google Scholar 

  10. Ginchev, I., Ivanov, V.I.: Second-order optimality conditions for problems with \(C^1\) data. J. Math. Anal. Appl. 340, 646–657 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gong, X.H.: Optimality conditions for efficient solution to the vector equilibrium problems with constraints. Taiwan. J. Math. 16, 1453–1473 (2012)

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Ivanov, V.I.: Second-order optimality conditions for inequality constrained problems with locally Lipschitz data. Optim. Lett. 4, 597–608 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Jiménez, B., Novo, V.: A finite dimensional extension of Lyusternik theorem with applications to multiobjective optimization. J. Math. Anal. Appl. 270, 340–356 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Luu, D.V.: Necessary and sufficient conditions for efficiency via convexificators. J. Optim. Theory Appl. 160, 510–526 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. 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 

  19. 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 

  20. Luu, D.V., Hang, D.D.: On optimality conditions for vector variational inequalities. J. Math. Anal. Appl. 412, 792–804 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Luu, D.V., Hang, D.D.: Efficient solutions and optimality conditions for vector equilibrium problems. Math. Methods Oper. Res. 79, 163–177 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Luu, D.V., Hang, D.D.: On efficiency conditions for nonsmooth vector equilibrium problems with equilibrium constraints. Numer. Funct. Anal. Optim. 36, 1622–1642 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ma, B.C., Gong, X.H.: Optimality conditions for vector equilibrium problems in normed spaces. Optimization 60, 1441–1455 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Morgan, J., Romaniello, M.: Scalarization and Kuhn–Tucker-like conditions for weak vector generalized quasivariational inequalities. J. Optim. Theory Appl. 130, 309–316 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Páles, Z., Zeidan, V.M.: Nonsmooth optimum problems with constraints. SIAM J. Control Optim. 32(5), 1476–1502 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pavel, N.H., Huang, J.K., Kim, J.K.: Higher order necessary conditions for optimization. Libertas Math. 14, 41–50 (1994)

    MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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Acknowledgements

The author is grateful to the referees for their valuable comments and suggestions which improve the paper. This research was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.01-2017.301

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  1. Institute of Mathematics, Vietnam Academy of Science and Technology, Thang Long University, Hanoi, Vietnam

    Do Van Luu

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  1. Do Van Luu
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Luu, D.V. Second-order necessary efficiency conditions for nonsmooth vector equilibrium problems. J Glob Optim 70, 437–453 (2018). https://doi.org/10.1007/s10898-017-0556-3

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  • DOI: https://doi.org/10.1007/s10898-017-0556-3

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