OPTIMALITY CONDITIONS AND DUALITY RESULTS FOR NONSMOOTH VECTOR OPTIMIZATION PROBLEMS WITH THE MULTIPLE INTERVAL-VALUED OBJECTIVE FUNCTION

doi:10.1016/S0252-9602(17)30062-0

Abstract

Abstract:

In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a (weakly) LU-efficient solution in the considered nonsmooth multiobjective programming problem with the multiple interval-objective function. Further, the sufficient optimality conditions for a (weakly) LU-efficient solution and several duality results in Mond-Weir sense are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem with the multiple intervalobjective function are convex.

Key words: nonsmooth multiobjective programming problem with the multiple intervalobjective function, Fritz John necessary optimality conditions, Karush-KuhnTucker necessary optimality conditions, (weakly) LU-efficient solution, MondWeir duality

Tadeusz ANTCZAK. OPTIMALITY CONDITIONS AND DUALITY RESULTS FOR NONSMOOTH VECTOR OPTIMIZATION PROBLEMS WITH THE MULTIPLE INTERVAL-VALUED OBJECTIVE FUNCTION[J].Acta mathematica scientia,Series B, 2017, 37(4): 1133-1150.

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References

[1] Alefeld G, Herzberger J. Introduction to Interval Computations. New York:Academic Press, 1983
[2] Antczak T. On nonsmooth (Φ, ρ)-invex multiobjective programming in finite-dimensional Euclidean spaces. J Adv Math Stud, 2014, 7:127-145
[3] Bhurjee A K, Panda G. Efficient solution of interval optimization problem. Math Method Oper Res, 2012, 76:273-288
[4] Bitran G R. Linear multiple objective problems with interval coefficients. Manage Sci, 1980, 26:694-706
[5] Chalco-Cano Y, Lodwick W A, Rufian-Lizana A. Optimality conditions of type KKT for optimization problem with interval-valued objective function via generalized derivative. Fuzzy Optim Decis Ma, 2013, 12:305-322
[6] Chanas S, Kuchta D. Multiobjective programming in optimization of interval objective functions-a generalized approach. Eur J Oper Res, 1996, 94:594-598
[7] Charnes A, Granot F, Phillips F. An algorithm for solving interval linear programming problems. Oper Res, 1977, 25:688-695
[8] Clarke F H. Optimization and Nonsmooth Analysis. John Wiley & Sons, Inc, 1983
[9] Hosseinzade E, Hassanpour H. The Karush-Kuhn-Tucker optimality conditions in interval-valued multiobjective programming problems. J Appl Math Inf, 2011, 29:1157-1165
[10] Ishihuchi H, Tanaka M. Multiobjective programming in optimization of the interval objective function. Eur J Oper Res, 1990, 48:219-225
[11] Jana M, Panda G. Solution of nonlinear interval vector optimization problem. Oper Res Int J, 2014, 14:71-85
[12] Jayswal A, Stancu-Minasian I, Ahmad I. On sufficiency and duality for a class of interval-valued programming problems. Appl Math Comput, 2011, 218:4119-4127
[13] Karmakar S, Bhunia A K. An alternative optimization technique for interval objective constrained optimization problems via multiobjective programming. J Egypt Math Soc, 2014, 22:292-303
[14] Mond B, Weir T. Generalized concavity and duality//Schaible S, Ziemba W T, eds. Generalized Concavity in Optimization and Economics. New York:Academic Press, 1981:263-279
[15] Preda V. Interval-valued optimization problems involving (α, ρ)-right upper-Dini-derivative functions. Sci World J, 2014, 2014:Article ID 750910
[16] Rockafellar R T. Convex Analysis. Princeton New Jersey:Princeton University Press, 1970
[17] Sun Y, Wang L. Optimality conditions and duality in nondifferentiable interval-valued programming. J Ind Manag Optim, 2013, 9:131-142
[18] Urli B, Nadeau R. An interactive method to multiobjective linear programming problems with interval coefficients. Infor, 1992, 30:127-137
[19] Wu H -C. The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function. Eur J Oper Res, 2007, 176:46-59
[20] Wu H -C. Wolfe duality for interval-valued optimization. J Optim Theory Appl, 2008, 138:497-509
[21] Wu H -C. On interval-valued nonlinear programming problems. J Math Anal Appl, 2008, 338:299-316
[22] Wu H -C. The Karush-Kuhn-Tucker optimality conditions in multiobjective programming problems with interval-valued objective functions. Eur J Oper Res, 2009, 196:49-60
[23] Wu H -C. Duality theory for optimization problems with interval-valued objective functions. J Optim Theory Appl, 2010, 144:615-628
[24] Zhou H -C, Wang Y -J. Optimality condition and mixed duality for interval-valued optimization//Fuzzy Information and Engineering, Advances in Intelligent and Soft Computing 62. Proceedings of the Third International Conference on Fuzzy Information and Engineering (ICFIE 2009), Springer, 2009:1315-1323