Continuous OptimizationSecond order symmetric duality in non-differentiable multiobjective programming with F-convexity☆
Introduction
In mathematical programming, a pair of primal and dual problems is called symmetric if the dual of the dual is the primal problem, that is, if the dual problem is expressed in the form of the primal problem, then its dual is the primal problem. However, the majority of dual formulations in nonlinear programming do not possess this property. The first symmetric dual formulation for quadratic programs was proposed by Dorn [5]. Subsequently, Dantzig et al. [4] and Mond [14] formulated a pair of symmetric dual programs for a real valued function f(x,y), that is convex in the first variable and concave in the second variable. Mond and Weir [17] then gave another pair of symmetric dual nonlinear programs in which a weaker convexity assumption was imposed on f. Later, Mond and Schechter [16] constructed two new symmetric dual pairs in which the objectives contain a support function and are therefore non-differentiable.
Weir and Mond [20] discussed symmetric duality in multiobjective programming by using the concept of efficiency. Chandra and Prasad [3] presented a pair of multiobjective programming problems by associating a vector valued infinite game to this pair. Kumar and Bhatia [12] discussed multiobjective symmetric duality by using a nonlinear vector valued function of two variables corresponding to various objectives. Gulati et al. [6] also established duality results for multiobjective symmetric dual problems without non-negativity constraints.
The study of second order duality is significant due to the computational advantage over first order duality as it provides tighter bounds for the value of the objective function when approximations are used (see [8], [13], [15]). Mangasarian [13] considered a nonlinear programming and discussed second order duality under an inclusion condition. Mond [15] was the first one to present second order symmetric dual models and proved second order symmetric duality theorems under second order convexity. Later, Jeyakumar [11] and Yang [21] also discussed second order Mangasarian type dual formulation under ρ-convexity and generalized representation conditions respectively. Bector and Chandra [1] studied Mond–Weir type second order primal and dual nonlinear programs and established second order symmetric duality results for these programs. Later on, Yang [22] generalized the results of Bector and Chandra [1] to nonlinear programs involving second order pseudo-invexity. Recently, Hou and Yang [10] extended the results in [22] to non-differentiable nonlinear programming problems under second order F-pseudo-convexity assumptions. More recently, Suneja et al. [19] presented a pair of Mond–Weir type multiobjective second order symmetric dual programs and gave their duality results. Yang et al. [23] introduced a pair of Wolfe type non-differentiable second order symmetric dual programs and established weak and strong duality theorems.
In this paper, motivated by Mond and Schechter [16], Hou and Yang [10] and Suneja et al. [19], we show that there is a new pair of second order symmetric models for a class of non-differentiable multiobjective programs. Weak duality, strong duality and converse duality theorems are established under F-convexity assumptions. Our study extends naturally and improves some of the known results in [1], [6], [10], [16], [19], [22].
Section snippets
Notations and preliminaries
Let be the n-dimensional Euclidean space and let be its non-negative orthant.
The following ordering relations for vectors in will be used in this paper:The negation of x≤y is denoted by x≰y.
Let f(x,y) be a real valued thrice continuously differentiable function defined on an open set in . Let denote the gradient vector of f with respect to x at . Also, let denote the Hessian matrix with respect to x
Mond–Weir type symmetric duality
We now state the following pair of second order Mond–Weir type non-differentiable multiobjective programming problem with k-objectives:
Primal problem (P)and
Dual problem (D)where
Remarks and example
Our results extend, unify and improve the works of Bector and Chandra [1], Gulati et al. [6], Hou and Yang [10], Mond and Schechter [16], Suneja et al. [19] and Yang [22].
- (i)
If C={0}, D={0}, k=1, then (P) and (D) are reduced to programs studied in [1], [22].
- (ii)
If C={0}, D={0}, then (P) and (D) are reduced to programs studied in [19].
- (iii)
If k=1, then (P) and (D) are reduced to programs studied in [10].
- (iv)
If p=q=0, k=1, then (P) and (D) become a pair of symmetric non-differentiable dual programs considered in
Acknowledgements
The authors are thankful to three anonymous reviewers for their many valuable comments on an early version of this paper.
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This research was partially supported by the National Natural Science Foundation of China, Trans-Century Training Programme Foundation for the Talents by Chinese Ministry of Education.