Multiobjective Fractional Symmetric Duality in Mathematical Programming with ( C , G f )-Invexity Assumptions

In this paper, a new class of (C, G f )-invex functions introduce and give nontrivial numerical examples which justify exist such type of functions. Also, we construct generalized convexity definitions (such as, (F, G f )-invexity, C-convex etc.). We consider Mond–Weir type fractional symmetric dual programs and derive duality results under (C, G f )-invexity assumptions. Our results generalize several known results in the literature.


Introduction
The goal of optimization is to find the best value for each variable in order to achieve satisfactory performance.Optimization is an active and fast growing research area and has a great impact on the real world.In most real life problems, decisions are made taking into account several conflicting criteria, rather than by optimizing a single objective.Such a problem is called multiobjective programming.Problems of multiobjective programming are widespread in mathematical modelling of real world systems problems for a very broad range of applications.
In 1981, Hanson [1] introduced the concept of invexity which is an extension of differentiable convex function and proved the sufficiency of Kuhn-Tucker conditions.Antczak [2] introduced the concept of G-invex functions and derived some optimality conditions for constrained optimization problems under G-invexity.In [3], Antczak extended the above notion by defining a vector valued G f -invex function and proved necessary and sufficient optimality conditions for a multiobjective nonlinear programming problem.Recently, Kang et al. [4] defined G-invexity for a locally Lipchitz function and obtained optimality conditions for multiobjective programming using these functions.Many researchers have worked related to the same area [5][6][7].
In the last several years, various optimality and duality results have been obtained for multiobjective fractional programming problems.Bector and Chandra Motivated by various concepts of generalized convexity.Ferrara and Stefaneseu [8] used the (φ, ρ)-invexity to discuss the optimality conditions and duality results for multiobjective programming problem.Further, Stefaneseu and Ferrara [9] introduced a new class of (φ, ρ) ω -invexity for a multiobjective program and established optimality conditions and duality theorems under these assumptions.
In this article, we have introduced various definitions (C, G f )-invexity/(F, G f )-invexity and constructed nontrivial numerical examples illustrates the existence of such functions.We considered a pair of multiobjective Mond-Weir type symmetric fractional primal-dual problems.Further, under the (C, G f )-invexity assumptions, we derive duality results.

Preliminaries and Definitions
Consider the following vector minimization problem: .., g m } : X → R m are differentiable functions defined on X.

Definition 1 ([10]
).A point x ∈ X 0 is said to be an efficient solution of (MP) if there exists no other x ∈ X 0 such that f r (x) < f r ( x), for some r = 1, 2, ..., k and f i (x) Then, the function C is said to be convex on R n with respect to third argument iff for any fixed (x, u) ∈ X × X, Now, we introduce the definition of C-convex function: Definition 3 ([12]).The function f is said to be C-convex at u ∈ X such that ∀x ∈ X, If the above inequality sign changes to ≤, then f is called C-concave at u ∈ X.
If the above inequality sign changes to ≤, then f is called G f -concave at u ∈ X. Definition 5. A functional F : X × X × R n → R is said to be sublinear with respect to the third variable if for all (x, u) ∈ X × X, , for all α ∈ R + and a ∈ R n .Now, we introduce the definition of a differentiable vector valued (F, G f )-invex function.
Definition 6.The function f is said to be (F, G f )-invex at u ∈ X if there exist sublinear functional F and a differentiable function Next, we introduce the definition of (C, G f )-invex function: If the above inequalities sign changes to ≤, then f is called (F, G f )-incave/ (F, G f )-pseudoincave at u ∈ X.
Definition 9. Let f : X → R k be a vector-valued differentiable function.If there exist convex function C and a differentiable function increasing on the range of I f i and a vector valued function η : Now, we give a nontrivial example which is (C, G f )-invex function, but on the either side the function f cannot hold the definitions like as (F, G f )-invex, F-convex and C-convex.
Now, we will show that f is (C, G f )-invex at u = 0.For this, we have to claim that Substituting the values of f 1 , f 2 , G f 1 and G f 2 in the above expressions, we obtain and which at u = 0 yield τ 1 = x 36 + x 28 + x 12 and τ 2 = 0.

This expression may not be non-negative for all
Finally, C x,u is not sublinear in its third position.Hence, function f is neither F nor (F, G f )-invex functions.

G-Mond-Weir Type Primal-Dual Model
In this section, we consider the following pair of multiobjective fractional symmetric primal-dual programs: G f i : I f i → R and G g i : I g i → R are differentiable strictly increasing functions on their domains.It is assumed that in the feasible regions, the numerators are nonnegative and denominators are positive.Now, Let U = (U 1 , U 2 , ..., U k ) and V = (V 1 , V 2 , ..., V k ).Then, we can express the programs (MFP) and (MFD) equivalently as: (MFD) V Minimize V subject to Next, we prove duality theorems for (MFP) U and (MFP) V , which one equally apply to (MFP) and (MFD), respectively.Theorem 1. (Weak duality).Let (x, y, U, λ) and (u, v, V, λ) be feasible for (MFP) U and (MFD) V , respectively.Let (i) f (., v) be (C, G f )-invex at u for fixed v, (ii) g(., v) be (C, G g )-incave at u for fixed v, (iii) f (x, .)be ( C, G f )-incave at y for fixed x, (iv) g(x, .)be ( C, G f )-invex at y for fixed x, Then, U V.

Proof.
By hypotheses (i) and (ii), we have and Using (v), λ > 0, λ i τ , and ) and ( 9)- (10), respectively, we obtain and Now, summing over i and adding the above two inequalities and using convexity of C x,u , we have Now, from (6), we have Hence, for this a, C x,u (a) ≥ −u T a ≥ 0 from (vii) .Using this in (11), we obtain Using (5) in above inequality, we get From hypotheses (iii) − (v) and from the condition (vii), Adding the inequalities ( 12) and ( 13), we get Since λ > 0 and using (vi), it follows that U V. This completes the proof.
Proof.The proof follows on the lines of Theorem 2.