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Karush–Kuhn–Tucker optimality conditions and duality for convex semi-infinite programming with multiple interval-valued objective functions

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Abstract

This paper deals with convex semi-infinite programming with multiple interval-valued objective functions. We first investigate necessary and sufficient Karush–Kuhn–Tucker optimality conditions for some types of optimal solutions. Then, we formulate types of Mond–Weir and Wolfe dual problems and explore duality relations under convexity assumptions. Some examples are provided to illustrate the advantages of our results in some cases.

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References

  1. Ahmad, I., Singh, D., Dar, B.A.: Optimality and duality in non-differentiable interval valued multiobjective programming. Int. J. Math. Oper. Res. 11, 332–356 (2017)

    Article  MathSciNet  Google Scholar 

  2. Antczak, T.: Optimality conditions and duality results for nonsmooth vector optimization problems with the multiple interval-valued objective function. Acta Math. Sci. 37, 1133–1150 (2017)

    Article  MathSciNet  Google Scholar 

  3. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  4. Bhurjee, A.K., Padhan, S.K.: Optimality conditions and duality results for non-differentiable interval optimization problems. J. Appl. Math. Comput. 50, 59–71 (2016)

    Article  MathSciNet  Google Scholar 

  5. Caristi, G., Ferrara, M.: Necessary conditions for nonsmooth multiobjective semi-infinite problems using Michel–Penot subdifferential. Decis. Econ. Finance 40, 103–113 (2017)

    Article  MathSciNet  Google Scholar 

  6. Chalco-Cano, Y., Lodwick, W.A., Osuna-Gómez, R., Rufián-Lizana, A.: The Karush–Kuhn–Tucker optimality conditions for fuzzy optimization problems. Fuzzy Optim. Decis. Mak. 15, 57–73 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Chuong, T.D., Yao, J.C.: Isolated and proper efficiencies in semi-infinite vector optimization problems. J. Optim. Theory Appl. 162, 447–462 (2014)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  10. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Springer, Berlin (1993)

    Book  Google Scholar 

  13. Jayswal, A., Ahmad, I., Banerjee, J.: Nonsmooth interval-valued optimization and saddle-point optimality criteria. Bull. Malays. Math. Sci. Soc. 39, 1391–1441 (2016)

    Article  MathSciNet  Google Scholar 

  14. Kabgani, A., Soleimani-damaneh, M.: Characterization of (weakly/properly/robust) efficient solutions in nonsmooth semi-infinite multiobjective optimization using convexificators. Optimization 67, 217–235 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Kumar, P., Sharma, B., Dagar, J.: Interval-valued programming problem with infinite constraints. J. Oper. Res. Soc. China 6, 611–626 (2018)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  18. Luu, D.V., Mai, T.T.: Optimality and duality in constrained interval-valued optimization. 4OR 16, 311–337 (2018)

    Article  MathSciNet  Google Scholar 

  19. Mond, B., Weir, T.: Generalized concavity and duality. In: Schaible, S., Ziemba, W.T. (eds.) Generalized Concavity in Optimization and Economics, pp. 263–279. Academic Press, New York (1981)

    MATH  Google Scholar 

  20. Osuna-Gómez, R., Hernádez-Jiménez, B., Chalco-Cano, Y., Ruiz-Gazón, G.: New efficiency conditions for multiobjective interval-valued programming problems. Inf. Sci. 420, 235–248 (2017)

    Article  Google Scholar 

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

    Book  Google Scholar 

  22. Singh, D., Dar, B.A., Kim, D.S.: KKT optimality conditions in interval-valued multiobjective programming with generalized differentiable functions. Eur. J. Oper. Res. 254, 29–39 (2016)

    Article  MathSciNet  Google Scholar 

  23. Sun, Y., Wang, L.: Optimality conditions and duality in nondifferentiable interval-valued programming. J. Ind. Manag. Optim. 9, 131–142 (2013)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. 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, Article ID 22 (2019)

  26. Vaz, A.I.F., Fernandes, E.M., Gomes, M.P.S.: Robot trajectory planning with semi-infinite programming. Eur. J. Oper. Res. 153, 607–617 (2004)

    Article  MathSciNet  Google Scholar 

  27. Vaz, A.I.F., Ferreira, E.C.: Air pollution control with semi-infinite programming. Appl. Math. Model. 33, 1957–1969 (2009)

    Article  MathSciNet  Google Scholar 

  28. Wolfe, P.: A duality theorem for nonlinear programming. Quart. Appl. Math. 19, 239–244 (1961)

    Article  MathSciNet  Google Scholar 

  29. Wu, H.C.: The Karush–Kuhn–Tucker optimality conditions in multiobjective programming problems with interval-valued objective functions. Eur. J. Oper. Res. 196, 49–60 (2009)

    Article  MathSciNet  Google Scholar 

  30. Wu, H.C.: The optimality conditions for optimization problems with convex constraints and multiple fuzzy-valued objective functions. Fuzzy Optim. Decis. Mak. 8, 295–321 (2009)

    Article  MathSciNet  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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  1. Department of Mathematics, College of Natural Sciences, Can Tho University, Can Tho, 900000, Vietnam

    Le Thanh Tung

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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. 62, 67–91 (2020). https://doi.org/10.1007/s12190-019-01274-x

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