Multiobjective fractional programming problems and the sufficient condition involving Hb – (p, r)-η- invex function

On the basis of arcwise connected convex functions and ) , ( η − r p invex functions,we established ) , ( Hb η − − r p invex functions.Based on the generalized invex assumption of new functions,the solutions of a class of multiobjective fractional programming problems are studied,and the sufficient optimality condition for the feasible solutions of multiobjective fractional programming problems to be efficient solutions are established and proved.


Introduction
Multiobjective programming is a branch of mathematical programming. The idea of multiobjective optimization was first put forward by French economist V. pareto in 1896. Multiobjective programming is widely used in many practical problems, such as economy, management, military affairs, science and engineering design. Convex function is one of the most widely used concepts in modern mathematics. In view of its importance in mathematical programming theory and application, many researches are devoted to popularizing these concepts to expand their application scope.M.A.Hanson studied the sufficiency of the Kuhn-Tucker [1];Avriel M and Zang I put forward a new class of generalized convex functions, and on this basis, gives some regularity conditions satisfying the characteristics of local-global minimum[2]; In reference[3], a new class of generalized convex functions is defined by means of symmetric gradient.In reference [4][5], the optimality conditions and Wolfe-type duality of two classes of invariant convex multiobjective nonlinear programming are proposed. In reference [6][7][8][9][10][11][12], the concepts of arc, arc connected set and arc connected convex function are established , and several new types of arc connected convex functions on this basis are defined; Tadeusz Antczak extended the generalized concepts of functions and sets and the convexity of functions, and defined invariant convex functions and invariant convex functions [13][14].
In this paper, a new class of generalized convex functions-invariant convex functions is defined on the basis of arc-wise connected convex functions and invariant convex functions. Based on the generalized invariant convex assumption of new functions, the solutions of a class of multiobjective fractional programming problems are studied, and some optimality sufficient conditions for the feasible solutions of multiobjective fractional programming problems to be efficient solutions are established and proved.

Notation and function definition
In this paper,we consider the following multiobjective fractional programming problem: be the set of all feasible solutions of (FP).
is an arc connected 1 x and 2 x in X .The function f is said to posses a right derivative,denoted by 13 The differentiable function invex with respect to the same η and b at each On the basis of arcwise connected convex functions and Definition2.6 Let n R X ∈ is an arc set,the differentiable real function f is said to be - invex with respect to η and b at X u ∈ on X if there exist a function invex with respect to the same η and b at each X u ∈ on X .

Definition2.7 Let
n R X ∈ is an arc set,the differentiable real function f is said to be invex with respect to the same η and b at each X u ∈ on X .
x with respect to η and b ; Then * x is an efficient solution for (FP).
Suppose contrary to the result that * x is not an efficient solution of (FP).Then there exists a feasible solution x of (FP) such that , ,..., 2 , 1 ) ( By (3.5) and the above inequalities,we have If * x is not an efficient solution of (FP),then we get (3.8) in the same way.But (3.8) contradicts (3.14).Therefore, * x is an efficient solution of (FP).This completes the proof.

Conclusion
In this paper,we established