Symmetric duality for multiobjective fractional variational problems involving cones

Abstract

In this paper, a pair of multiobjective fractional variational symmetric dual problems over cones is formulated. Weak, strong and converse duality theorems are established under generalized $\mathcal{F}$-convexity assumptions. Moreover, self duality theorem is also discussed.

Introduction

The notion of symmetric duality in nonlinear programming, in which the dual of the dual is the primal, was first introduced by Dorn [13], but significantly developed and studied by Dantzig et al. [11], Mond and Weir [23], and Chandra et al. [7]. Bazaraa and Goode [4] generalized the results of Dantzig et al. [11] to arbitrary cones. Nanda and Das [24] studied symmetric duality in fractional programming involving arbitrary cones assuming the functions to be pseudoinvex. Chandra and Kumar [10] pointed out some logical shortcomings in the proofs of duality theorems of Nanda and Das [24]. Suneja et al. [26] formulated a pair of multiobjective symmetric dual programs over arbitrary cones and proved various duality results for cone–convex functions. Recently, Khurana [19] discussed multiobjective symmetric duality results for Mond–Weir type problems under generalized cone–invex functions.

Mond and Hanson [22] and Bector et al. [5] extended symmetric duality to variational programming, giving continuous analogous of the results of Dantzig et al. [11] and Mond and Weir [23], respectively. Smart and Mond [25] studied symmetric duality for variational problems with invexity, omitting the nonnegativity constraints taken by Mond and Hanson [22]. Gulati et al. [16] presented a pair of multiobjective symmetric dual variational problems and discussed duality results under generalized invexity. In [15], Gulati et al. generalized the results of Mond and Hanson [22] and Bector et al. [5] by constraining some of the primal and dual variables to belong to arbitrary sets of integers. Recently, Ahmad and Husain [3] formulated minimax mixed integer multiobjective symmetric dual variational programs over cones and obtained appropriate duality results.

Chandra and Husain [9] studied symmetric duality for fractional variational problems. In [17], Gulati et al. established usual duality results for static and continuous symmetric dual fractional programming problems without nonnegativity constraints. Recently, Kim et al. [20] and Ahmad [1] discussed symmetric duality results for multiobjective fractional variational programs under invexity and pseudoinvexity, respectively.

Generalizing convex functions, Hanson and Mond [18] introduced functions which satisfy certain convexity type properties with sublinear functionals. Egudo and Mond [14] named these functions as $\mathcal{F}$-convex, $\mathcal{F}$-pseudoconvex, and $\mathcal{F}$-quasiconvex functions. Examples of these functions have been given in [14], [18] also. Later on, Chandra et al. [8] used these definitions in another form to discuss symmetric duality. Motivated by Hanson and Mond [18], Egudo and Mond [14], and Chandra et al. [8], we propose the continuous version of generalized $\mathcal{F}$-convexity, and use this concept to prove symmetric duality results for multiobjective fractional variational symmetric problems involving arbitrary cones. At the end, self duality theorem is also proved.