# Duality for the class of a multiobjective problem with support functions under $K$-$G_f$-invexity assumptions

Document Type: Research Paper

Authors

Department of Mathematics‎, ‎Indian Institute of Technology Roorkee‎, ‎Roorkee 247 667‎, ‎India.

Abstract

‎In this article‎, ‎we formulate two dual models Wolfe and Mond-Weir related to symmetric nondifferentiable multiobjective programming problems‎. ‎Furthermore‎, ‎weak‎, ‎strong and converse duality results are established under $K$-$G_f$-invexity assumptions‎. ‎Nontrivial examples have also been depicted to illustrate the theorems obtained in the paper‎. ‎Results established in this paper unify and extend some previously known results appeared in the literature

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Main Subjects

### References

R.P. Agarwal, I. Ahmad and S.K. Gupta, A note on higher-order nondifferentiable symmetric duality in multiobjective programming, Appl. Math. Lett. 24 (2011), no. 8, 1308--1311.

T. Antczak, On G-invex multiobjective programming, Part I. Optimality, J. Global Optim. 43 (2009), no. 1, 97--109.

T. Antczak, Saddle point criteria and Wolfe duality in nonsmooth (ϕ; ρ)-invex vector optimization problems with inequality and equality constraints, Int. J. Comput. Math. 92 (2015), no. 5, 882--907.

M.S. Bazaraa and J.J. Goode, On symmetric duality in nonlinear programming, Oper. Res. 21 (1973), no. 1, 1--9.

X. Chen, Higher-order symmetric duality in nondifferentiable multiobjective programming problems, J. Math. Anal. Appl. 290 (2004), no. 2, 423--435.

Y. Dehui and L. Xiaoling, Multiobjective program with support functions under (G;C; ρ)-convexity assumptions, J. Syst. Sci. Complex. 28 (2015), no. 5, 1148--1163.

W.S. Dorn, A symmetric dual theorem for quadratic programs, J. Oper. Res. Soc. Japan 2 (1960), no. 3, 93--97.

T.R. Gulati and G. Mehndiratta, Nondifferentiable multiobjective Mond-Weir type second-order symmetric duality over cones, Optim. Lett. 4 (2010), no. 2, 293--309.

S.K. Gupta and A. Jayswal, Multiobjective higher-order symmetric duality involving generalized cone-invex functions, Comput. Math. Appl. 60 (2010), no. 12, 3187--3192.

S.K. Gupta, N. Kailey and S. Kumar, Duality for nondifferentiable multiobjective higher-order symmetric programs over cones involving generalized (F; α;ρ;d)-convexity, J. Inequal. Appl. 298 (2012) 16 pages.

A. Jayswal and K. Kummari, Higher-order duality for multiobjective programming problem involving (ϕ;ρ)-invex functions, J. Egyptian Math. Soc. 23 (2015), no. 1, 12--19.

A. Jayswal and K. Kummari, On nondifferentiable minimax semi-infinite programming problems in complex spaces, Georgian Math. J. 23 (2016), no. 3, 367--380.

H. Jiao, Sufficiency and duality for a nonsmooth vector optimization problem with generalized α-dI -type-I univexity over cones, Bull. Iranian Math. Soc. 42 (2016), no. 2, 285--295.

S. Khurana, Symmetric duality in multiobjective programming involving generalized cone-invex functions, European J. Oper. Res. 165 (2005), no. 1, 20--26.

K.M. Miettinen, Nonlinear Multiobjective Optimization, Kluwer, Boston, 1999.

P.M. Pardalos, Y. Siskos and C. Zopounidis, Advances in Multicriteria Analysis, Kluwer, Netherlands, 1995.

M. Soleimani-damaneh, On optimality and duality for multipleobjective optimization under generalized type I univexity, Int. J. Comput. Math. 86 (2009), no. 8, 1345--1354.

S.K. Suneja, S. Agarwal and S. Davar, Multiobjective symmetric duality involving cones, European J. Oper. Res. 141 (2002), no. 3, 471--479.

S.K. Suneja and P. Louhan, Higher-order symmetric duality under cone-invexity and other related concepts, J. Comput. Appl. Math. 255 (2014), 825--836.

L. Xianjun, Sufficiency and duality for nonsmooth multiobjective programming problems involving generalized univex functions, J. Syst. Sci. Complex. 26 (2013), no. 6, 1002--1018.

V.I. Zorkaltsev, Symmetric duality in optimization and applications (Russian), Proceedings of Higher Education Institutes. Mathematics 2 (2006) 53--59.

### History

• Receive Date: 05 July 2016
• Revise Date: 31 January 2017
• Accept Date: 31 January 2017