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

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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

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

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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 

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

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  • 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 

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

    Article  MathSciNet  Google Scholar 

  • 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 

  • 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • 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 

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

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • 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 

  • 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 

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

    Book  Google Scholar 

  • 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 

  • 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

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

  • 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)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  • 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 

  • 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)

  • 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

  • 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

  • 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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  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. 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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