Advertisement

Optimality and Duality in Nonsmooth Conic Vector Optimization

  • Thai Doan ChuongEmail author
Regular Paper
  • 33 Downloads

Abstract

This article is concerned with a nonsmooth vector optimization problem involving conic constraints. We employ some advanced tools of variational analysis and generalized differentiation to establish necessary conditions for (weakly) efficient solutions of the conic vector optimization problem, where the fuzzy necessary condition and sequential necessary condition are expressed in terms of the Fréchet subdifferential and the exact necessary condition is in terms of the limiting/Mordukhovich subdifferential of the related functions. Sufficient conditions for (weakly) efficient solutions of the underlying problem are also provided by means of introducing the concepts of (strictly) generalized convex vector functions with respect to a cone. In addition, we propose a dual problem to the conic vector optimization problem and explore weak, strong, and converse duality relations between these two problems.

Keywords

Necessary/sufficient conditions Duality Vector optimization Conic constraints Limiting/Mordukhovich subdifferential 

Mathematics Subject Classification

49K99 65K10 90C29 90C46 

Notes

Acknowledgements

The author would like to thank three anonymous referees for the valuable comments and suggestions.

References

  1. 1.
    Jahn, J.: Vector Optimization. Theory, Applications, and Extensions. Springer, Berlin (2004)zbMATHGoogle Scholar
  2. 2.
    Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin (1989)Google Scholar
  3. 3.
    Bot, R.I., Grad, S.-M.: Duality for vector optimization problems via a general scalarization. Optimization 60, 1269–1290 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ginchev, I., Guerraggio, A., Rocca, M.: Dini set-valued directional derivative in locally Lipschitz vector optimization. J. Optim. Theory Appl. 143, 87–105 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Jimenez, B., Novo, V.: First order optimality conditions in vector optimization involving stable functions. Optimization 57, 449–471 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
  7. 7.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  8. 8.
    Antczak, T.: Proper efficiency conditions and duality results for nonsmooth vector optimization in Banach spaces under \((\Phi,\rho )\)-invexity. Nonlinear Anal. 75, 3107–3121 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Brandao, A.J.V., Rojas-Medar, M.A., Silva, G.N.: Optimality conditions for Pareto nonsmooth nonconvex programming in Banach spaces. J. Optim. Theory Appl. 103, 65–73 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, J.W., Cho, Y.J., Kim, J.K., Li, J.: Multiobjective optimization problems with modified objective functions and cone constraints and applications. J. Global Optim. 49, 137–147 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fulga, C., Preda, V.: On optimality conditions for multiobjective optimization problems in topological vector space. J. Math. Anal. Appl. 334, 123–131 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hu, Y., Ling, C.: The generalized optimality conditions of multiobjective programming problem in topological vector space. J. Math. Anal. Appl. 290, 363–372 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pappalardo, M., Stocklin, W.: Necessary optimality conditions in nondifferentiable vector optimization. Optimization 50, 233–251 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory. Springer, Berlin (2006)CrossRefGoogle Scholar
  15. 15.
    Chuong, T.D., Kim, D.S.: Optimality conditions and duality in nonsmooth multiobjective optimization problems. Ann. Oper. Res. 217, 117–136 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chuong, T.D., Kim, D.S.: Nonsmooth semi-infinite multiobjective optimization problems. J. Optim. Theory Appl. 160, 748–762 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mordukhovich, B.S., Treiman, J.S., Zhu, Q.J.: An extended extremal principle with applications to set-valued optimization. SIAM J. Optim. 14, 359–379 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bao, T.Q., Mordukhovich, B.S.: Necessary conditions for super minimizers in constrained multiobjective optimization. J. Global Optim. 43, 533–552 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers in multiobjective optimization: existence and optimality conditions. Math. Program. 122, 301–347 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Chuong, T.D.: Linear matrix inequality conditions and duality for a class of robust multiobjective convex polynomial programs. SIAM J. Optim. 28, 2466–2488 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming. Optimization 55, 685–708 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Mordukhovich, B.S.: Variational Analysis and Applications. Springer, Cham (2018)CrossRefzbMATHGoogle Scholar
  23. 23.
    Mordukhovich, B.S., Nghia, T.T.A.: Nonsmooth cone-constrained optimization with applications to semi-infinite programming. Math. Oper. Res. 39, 301–324 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Jeyakumar, V., Lee, G.M., Dinh, N.: New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs. SIAM J. Optim. 14, 534–547 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Bot, R.I., Csetnek, E.R., Wanka, G.: Sequential optimality conditions for composed convex optimization problems. J. Math. Anal. Appl. 342, 1015–1025 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wolfe, P.: A duality theorem for nonlinear programming. Quart. Appl. Math. 19, 239–244 (1961)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Optimization and Applications Research GroupTon Duc Thang UniversityHo Chi Minh CityVietnam
  2. 2.Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam

Personalised recommendations