On Nondifferentiable Higher-Order Symmetric Duality in Multiobjective Programming Involving Cones
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On Nondifferentiable Higher-Order Symmetric Duality in Multiobjective Programming Involving Cones Xin Min Yang · Jin Yang · Tsz Leung Yip
Received: 31 October 2013 / Accepted: 25 November 2013 / Published online: 10 December 2013 © Operations Research Society of China, Periodicals Agency of Shanghai University, and Springer-Verlag Berlin Heidelberg 2013
Abstract In this paper, we point out some deficiencies in a recent paper (Lee and Kim in J. Nonlinear Convex Anal. 13:599–614, 2012), and we establish strong duality and converse duality theorems for two types of nondifferentiable higher-order symmetric duals multiobjective programming involving cones. Keywords Multiobjective programming · Higher-order Mond–Weir symmetric dual model · Higher-order Wolfe symmetric dual model · Strong duality theorems · Converse duality theorems · Nondifferentiability 1 Introduction Mangasarian [4] first formulated a class of higher-order dual problems for nonlinear programming problems. Later, in [5], Mond and Weir gave the conditions for duality and considered other higher-order duals. Mond and Zhang [6] obtained duality results for various higher-order dual problems under higher-order invexity assumptions. Higher-order duality in nonlinear programming has been studied in the This work was partially supported by the National Natural Science Foundation of China (Nos. 11271391 and 10831009) and the Natural Science Foundation of Chongqing (CSTC, No. 2011BA0030).
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X.M. Yang ( ) Department of Mathematics, Chongqing Normal University, Chongqing 400047, China e-mail: [email protected] J. Yang Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China T.L. Yip Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China
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last few years by many researchers. One practical advantage of higher-order duality is that it provides tighter bounds for the value of objective function of the primal problem when approximations are used because there are more parameters involved. Mishra and Rueda [8] considered higher-order duality for the nondifferentiable mathematical programming. They formulated a number of higher-order duals to a nondifferentiable programming problems and established duality under the higher-order generalized invexity conditions introduced in [7]. In [9], Yang et al. extended the results in [8] to a class of nondifferentiable multiobjective programs. Chen [2] studied higher-order symmetric duality for multiobjective nondifferentiable programs by introducing higher-order F-convexity. Agarwal et al. [1] extended the results of [2] to arbitrary cones and proved appropriate duality relations under higher-order K-Fconvexity assumptions. Recently, Lee and Kim [3] have presented higher-order symmetric dual programs for multiobjective problems. In the literature strong and converse duality theorems have been established assuming conditions on known quantities. However, in strong and converse duality theorem
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