Special class of G-Wolfe type fractional symmetric duality theorems under G-pseodoinvexity assumptions

In this article, a pair of G-Wolfe-type fractional programming problems is formulated. For a differentiable function, we consider the definitions of G-invexity/G-psedoinvexity, which extends some kinds of generalized convexity assumptions. In the next section, we prove the weak, strong and converse duality theorems under G-invexity/G-psedoinvexity assumptions.


Introduction
Duality assumes a significant part in optimization problems and is extremely valuable both hypothetically and in real life. If dual of dual is a primal problem , pair of primal and dual is said to have symmetric property. As opposed to linear programming, the vast majority dual formulation in nonlinear programming do not have the symmetric property. Various authors have considered fractional programming problems containing square root of positive semidefinite quadratic forms like Mond [1] and Zhang and Mond [2].
Convexity suspicions are regularly not fulfilled in genuine problems, so there was a need to debilitate them. One of the ways was the presentation of speculation of convexity specifically quasiconvexity and pseudoconvexity. One of the ways was the introduction of generalization of convexity namely quasi/pseudo-convexity. For more data on fractional programming, readers are advised to see [7,8,9,10,11,12].
In this paper, we consider G-invexity/pseudoinvexity assumptions. Generalized G-Wolfe type fractional symmetric dual is proposed and duality results are derived by using the above mentioned functions.

Preliminaries and Definitions
Let S 1 ⊆ R n and S 2 ⊆ R m be open sets and f (x, y) be real valued differentiable function defined on S 1 × S 2 . Let G : R → R be strictly increasing function in their range G : Remark 2.1. If the above inequality sign changes ≤, then the function f (x, y) is Gpseudoincave in the first variable at u ∈ S 1 for fixed v ∈ S 2 with respect to η 1 .
Definition 2.2. The function f (x, y) is G-pseudoinvex in the second variable at v ∈ S 2 for fixed u ∈ S 1 with respect to η 2 , such that for y ∈ S 2 , we have Remark 2.2. If the above inequality sign changes ≤, then the function f (x, y) is Gpseudoincave in the second variable at v ∈ S 2 for fixed u ∈ S 1 with respect to η 2 .
Remark 2.3. If the above inequality sign changes ≤, then the function f (x, y) is G -incave in the first variable at u ∈ S 1 for fixed v ∈ S 2 with respect to η 1 .
Remark 2.4. If the above inequality sign changes ≤, then the function f (x, y) is G -incave in the second variable at v ∈ S 2 for fixed u ∈ S 1 with respect to η 2 .

G-Wolfe type fractional symmetric pair of primal-dual model
In the following section, we formulate the following pair of G-Wolfe type fractional symmetric dual programming problem: Primal Problem (FWP): Min

Dual Problem (FWD):
Max The above primal-dual modls can be re-written as: v ≥ 0.
Let R 0 and S 0 be the sets of feasible solution of (EFWP) and (EFWD), respectively.
From assumption (iv), above inequality gives Hence, completes the result.
Remark 3.1 Since every invex function is pseudoinvex. Therefore above weak duality can also be obtained under G-pseudoinvex assumptions. (ii) f (x, .) be G-pseudoincave and g(x, .) be G-pseudoinvex at y for fixed x with respect to η 2 , Then, w ≥ t.
Next, we have to prove that the objective values of the problem are same. For this, it is sufficient to show that Now, multiplying (23), byx T and using (20), we havē Using (22) and (18), we obtainȳ Equations (24) and (25) givē i.e.
Then, there exists (v,w,t) ∈ R 0 and objective values are equal. Moreover, if all the hypotheses of weak duality theorem are satisfied, then (v,w,t) is an optimal solution of (EFWP).
Proof: Proof of strict converse duality theorem follows on the lines of Theorem 3.3, due to symmetric programming problem.

Conclusion
In this article, we considered fractional dual symmetric programming problem and derived duality theorems under G-invexity/G-pseudoinvexity conditions. The present work can be extended to multiobjective symmetric fractional dual programs. This may be taken as the future task of the researchers.