Strong Karush–Kuhn–Tucker Optimality Conditions for Borwein Properly Efficient Solutions of Multiobjective Semi-infinite Programming

Abstract

In this paper, we study strong Karush–Kuhn–Tucker optimality conditions for Borwein properly efficient solutions of multiobjective semi-infinite programming. By using some regularity conditions in the sense of Mordukhovich subdifferentials and Clarke subdifferentials, we obtain some strong necessary optimality conditions for Borwein properly efficient solutions. Some examples are also provided to illustrate that our regularity conditions have more advantages than the previous regularity conditions in some cases.

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References

  1. Borwein, J.M.: Proper efficient points for maximizations with respect to cones. SIAM J. Control Optim. 15, 57–63 (1977)

    MathSciNet  Article  Google Scholar 

  2. Burachik, R.S., Rizvi, M.M.: On weak and strong Kuhn–Tucker conditions for smooth multiobjective optimization. J. Optim. Theory Appl. 155, 477–491 (2012)

    MathSciNet  Article  Google Scholar 

  3. Burachik, R.S., Rizvi, M.M.: Proper effciency and proper Karush–Kuhn–Tucker conditions for smooth multiobjective optimization problems. Vietnam J. Math. 42, 521–531 (2014)

    MathSciNet  Article  Google Scholar 

  4. Caristi, G., Kanzi, N.: Karush–Kuhn–Tucker type conditions for optimality of nonsmooth multiobjective semi-infinite programming. Int. J. Math. Anal. 9, 1929–1938 (2015)

    Article  Google Scholar 

  5. Chuong, T.D., Kim, D.S.: Nonsmooth semi-infinite multiobjective optimization problems. J. Optim. Theory Appl. 160, 748–762 (2014)

    MathSciNet  Article  Google Scholar 

  6. Chuong, T.D., Kim, D.S.: Normal regularity for the feasible set of semi-infinite multiobjective optimization problems with applications. Ann. Oper. Res. 267, 81–99 (2018)

    MathSciNet  Article  Google Scholar 

  7. Chuong, T.D., Yao, J.-C.: Isolated and proper effciencies in semi-infinite vector optimization problems. J. Optim. Theory Appl. 162, 477–462 (2014)

    Article  Google Scholar 

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

    Google Scholar 

  9. Constantin, E.: First-order necessary conditions in locally Lipschitz multiobjective optimization. Optimization 67, 1447–1460 (2018)

    MathSciNet  Article  Google Scholar 

  10. Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)

    Google Scholar 

  11. Goberna, M.A., Kanzi, N.: Optimality conditions in convex multiobjective SIP. Math. Program. 164, 67–191 (2017)

    MathSciNet  Article  Google Scholar 

  12. Goberna, M.A., Lopéz, M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)

    Google Scholar 

  13. Goberna, M.A., Guerra-Vázquez, F., Todorov, M.I.: Constraint qualifications in convex vector semi-infinite optimization. Eur. J. Oper. Res. 249, 32–40 (2016)

    MathSciNet  Article  Google Scholar 

  14. Guerragio, A., Molho, E., Zaffaroni, A.: On the notion of proper efficiency in vector optimization. J. Optim. Theory Appl. 82, 1–21 (1994)

    MathSciNet  Article  Google Scholar 

  15. Ha, T.D.X.: The Fermat rule and Lagrange multiplier rule for various efficient solutions of set-valued optimization problems expressed in terms of coderivatives. In: Ansari, Q.H., Yao, J.-C. (eds.) Recent Advances in Vector Optimization, Chapter 12, pp. 417–466. Springer, Berlin (2011)

    Google Scholar 

  16. Kabgani, A., Soleimani-damaneh, M.: Characterization of (weakly/properly/robust) efficient solutions in nonsmooth semi-infinite multiobjective optimization using convexificators. Optimization 67, 217–235 (2017)

    MathSciNet  Article  Google Scholar 

  17. Kanzi, N.: Lagrange multiplier rules for non-differentiable DC generalized semi-infinite programming problems. J. Glob. Optim. 56, 417–430 (2013)

    MathSciNet  Article  Google Scholar 

  18. Kanzi, N.: Two constraint qualifications for non-differentiable semi-infinite programming problems using Fréchet and Mordukhovich subdifferentials. J. Math. Ext. 8, 83–94 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Kanzi, N.: On strong KKT optimality conditions for multiobjective semi-infinite programming problems with Lipschitzian data. Optim. Lett. 9, 1121–1129 (2015)

    MathSciNet  Article  Google Scholar 

  20. Kanzi, N.: Necessary and sufficient conditions for (weakly) efficient of non-differentiable multi-objective semi-infinite programming problems. Iran. J. Sci. Technol. Trans. A Sci 42, 1537–1544 (2018)

    MathSciNet  Article  Google Scholar 

  21. Kanzi, N., Nobakhtian, S.: Optimality conditions for nonsmooth semi-infinite multiobjective programming. Optim. Lett. 8, 1517–1528 (2014)

    MathSciNet  Article  Google Scholar 

  22. Liu, J.J., Zhao, K.Q., Yang, X.M.: Optimality and regularity conditions using Mordukhovich’s subdifferential. Pac. J. Optim. 13, 43–53 (2017)

    MathSciNet  MATH  Google Scholar 

  23. Long, X.J., Peng, Z.Y., Wang, X.F.: Characterizations of the solution set for nonconvex semi-infinite programming problems. J. Nonlinear Convex Anal. 17, 251–265 (2016)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  25. Luu, D.V.: Necessary efficiency conditions for vector equilibrium problems with general inequality constraints via convexificators. Bull. Braz. Math. Soc. New Ser. (2018). https://doi.org/10.1007/s00574-018-00124-x

    Article  MATH  Google Scholar 

  26. Luu, D.V., Mai, T.T.: On optimality conditions for Henig efficiency and superefficiency in vector equilibrium problems. Numer. Funct. Anal. Optim. 39, 1833–1854 (2018)

    MathSciNet  Article  Google Scholar 

  27. Maeda, T.: Constraint qualifications in multiobjective optimization problems: differentiable case. J. Optim. Theory Appl. 80, 483–500 (1994)

    MathSciNet  Article  Google Scholar 

  28. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, vol. I. Basic Theory. Springer, Berlin (2006a)

  29. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, vol. II. Applications. Springer, Berlin (2006b)

  30. Pandey, Y., Mishra, S.K.: On strong KKT type sufficient optimality conditions for nonsmooth multiobjective semi-infinite mathematical programming problems with equilibrium constraints. Oper. Res. Lett. 44, 148–151 (2016)

    MathSciNet  Article  Google Scholar 

  31. Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton (1970)

    Google Scholar 

  32. Tung, L.T.: Strong Karush–Kuhn–Tucker optimality conditions for multiobjective semi-infinite programming via tangential subdifferential. RAIRO Oper. Res. 52, 1019–1041 (2018)

    MathSciNet  Article  Google Scholar 

  33. Tung, L.T.: Karush-Kuhn-Tucker optimality conditions and duality for semi-infinite programming with multiple interval-valued objective functions. J. Nonlinear Funct. Anal. 2019, 1–21 (2019a)

  34. Tung, L.T.: Karush–Kuhn–Tucker optimality conditions and duality for convex semi-infinite programming with multiple interval-valued objective functions. J. Appl. Math. Comput. (2019b). https://doi.org/10.1007/s12190-019-01274-x

  35. Tung, L.T.: Karush–Kuhn–Tucker optimality conditions and duality for multiobjective semi-infinite programming via tangential subdifferentials. Numer. Funct. Anal. Optim. (2019c). https://doi.org/10.1080/01630563.2019.1667826

  36. Zhao, K.Q.: Strong Kuhn–Tucker optimality in nonsmooth multiobjective optimization problems. Pac. J. Optim. 11, 483–494 (2015)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to thank the Editors for the help in the processing of the article. The author is very grateful to the Anonymous Referee for the valuable remarks, which helped to improve the paper. This work is partially supported by Can Tho University.

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Correspondence to Le Thanh Tung.

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Tung, L.T. Strong Karush–Kuhn–Tucker Optimality Conditions for Borwein Properly Efficient Solutions of Multiobjective Semi-infinite Programming. Bull Braz Math Soc, New Series 52, 1–22 (2021). https://doi.org/10.1007/s00574-019-00190-9

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Keywords

  • Multiobjective semi-infinite programming
  • Mordukhovich subdifferentials
  • Clarke subdifferentials
  • Borwein properly efficient solutions
  • Strong KKT optimality conditions

Mathematics Subject Classification

  • 90C46
  • 90C29
  • 90C34
  • 49J52