Second order multiobjective mixed symmetric duality containing square root term with generalized invex function
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cond order multiobjective mixed symmetric duality containing square root term with generalized invex function A. K. Tripathy & G. Devi
Accepted: 3 September 2012 / Published online: 5 October 2012 # Operational Research Society of India 2012
Abstract In this paper we presented a pair of second order mixed symmetric dual for a class of nondifferentiable multiobjective programming involving square root term like (xTAx)1/2. We established weak duality, strong duality and converse duality theorems with their proofs under second order (Φ, ρ)-invexity and (Φ, ρ)-pseudo invexity assumption. Discussion on some particular cases shows that our results generalize earlier results in related domain. Keywords Second order (Φ, ρ)-invexity . Second order (Φ, ρ)-pseudo-invexity . Second order mixed symmetric dual . Schwartz inequality . Efficient solution
1 Introduction Duality theory has played an important role in the development of optimization theory. For nonlinear programming problems a number of duals have been suggested, among which the Wolfe dual proposed by Dorn [14] is well known. Symmetric duality in nonlinear programming was introduced in [14] by defining a symmetric dual problem for quadratic programs. Subsequently Dantzing et al. [12], Bazarra et al. [6] established symmetric duality results for convex/concave functions. Mond [26] established symmetric duality for nonlinear programming problem. Devi [13], Weir and Mond [30], Mond and Schechter [27] studied non differentiable symmetric duality for a class of optimization problem in which the objective function consist of support function. Husain et al. [17] formulate a pair of Mond Weir type second order symmetric dual and establish the duality results under pseudo convexity A. K. Tripathy (*) Department of Mathematics, Trident Academy of Technology, F2/A, Chandaka Industrial Estate, In front of Infocity, Patia, Bhubaneswar 751024 Odisha, India e-mail: [email protected] G. Devi Department of Computer Science, Ajayabinayak Institute of Technology, Cuttack, Odisha, India
OPSEARCH (Apr–Jun 2013) 50(2):260–281
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–pseudo concavity assumption. Ahmad and Husain [3, 5] presented non differentiable second order duality in multiobjective programming. Khurana [20] established symmetric duality in multiobjective programming under generalized cone-invex function. Kim [21] presented a nondifferentiable symmetric dual for multiobjective programming with cone constraint. In recent years, several extension and generalization have been considered for classical convexity. A significant generalization of convex function is that of in-vex function introduced by Hanson [15] and Craven [11]. After the work of Hanson and Craven, other types of differentiable functions have been introduced with the intent of generalizing in vex function from different point of view. Hanson and Mond [16] introduced the concept of F-convexity. The concept of generalized (F, ρ) convexity is introduced by Preda [28]. The (F, ρ) convexity was recently generalized to (∅,ρ) invexity by Caristi,Fer
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