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

**Keywords**

**Main Subjects**

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*-*d**I *-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.

November and December 2017

Pages 2233-2258

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