DUALITY IN MULTIOBJECTIVE FRACTIONAL PROGRAMMING PROBLEMS INVOLVING (H_p, r)-INVEX FUNCTIONS

Abstract

In this paper, we have taken step in the direction to establish weak, strong and strict converse duality theorems for three types of dual models related to multiojective fractional programming problems involving ($H_p$, r)-invex functions.

1. Introduction

Generalized convexity plays an important role in many aspects of optimization, such as optimality conditions, duality theorems, variational inequalities, saddle point theory and convergence of optimization algorithms, so the research on generalized convexity is one of the important aspects of mathematical programming problems.

The problem in which objective functions are ratio of two functions are termed as fractional programming problems. Such problems are studied in various fields like economics [3], information theory [12], heat exchange networking [24] and others. Duality in multiobjective fractional programming problems involving generalized convex functions have been of much interest in recent past, (see [4,5,8,14,16,18,22]) and the references cited therein. For more information about fractional programming problems, the reader may consult the research bibliography compiled by Stancu-Minasian [19,20,21].

Mukherjee [13] considered a multiobjective fractional programming problem and discussed the Mond-Weir type duality results under generalized convexity. Gulati and Ahmad [6] proved the duality results using Fritz John conditions for multiobjective programming problem involving generalized convex functions. Kaul et al. [9] derived duality results for a Mond-Weir type dual problem related to multiobjective fractional programming problem involving pseudolinear and η-pseudolinear functions. Osuna-Gómez et al. [15] focus his study to establish the optimality condition and duality theorems for a class of multiobjective fractional programs under generalized convexity assumptions by applying parametric approach.

The notion of convexity was not enough to meet the challenging demand of some problems on Economics and Engineering. To meet this demand the notion of invexity was introduced by Hanson [7] by substituting the linear term (x−y) appearing in the definition of convex functions with an arbitrary vector valued function.

Antczak [2] introduced a new class of functions named (p, r)-invex function, which is an extension of invex function. Recently, Jayswal et al. [8] focus his study on multiobjective fractional programming problems and derived sufficient optimality conditions and duality theorems involving (p, r) − ρ − (η, θ)-invex functions [11].

Yuan et al. [23] introduced new types of generalized convex functions and sets, which are called locally (Hp, r, α)−pre-invex and locally Hp-invex sets. They obtained also optimality conditions and duality theorems for a scalar nonlinear programming problem. Recently, Liu et al. [10] proposed the concept of (Hp, r)-invex function and focus his study to discuss sufficient optimality conditions to multiple objective programming problem and multiobjective fractional programming problem involving the aforesaid class of functions but no step was taken to prove the duality results involving (Hp, r)-invex functions.

In this paper, viewing the importance of duality theorems in optimization theory, we establish weak, strong and strict converse duality theorems involving (Hp, r)-invex function to three types of dual models related to mulitiobjective fractional programming problems. The organization of the remainder of this paper is as follows. The formulation of multiobjective fractional programming problem along with some definitions and notations related to (Hp, r)-invexity is given in Section 2. Weak, strong and strict converse duality theorems for three types of dual models related to multiobjective fractional programming problem under (Hp, r)-invexity are derived in Section 3 to Section 5. Finally, conclusions and further developments are given in Section 6.

 

2. Notation and Preliminaries

Throughout the paper, let Rn be the n-dimensional Euclidean space, Let x, y ∈ Rn. Then and x ≠ y.

Definition 2.1 ([2]). Let a1, a2 > 0, λ ∈ (0, 1) and r ∈ R. Then the weighted r-mean of a1 and a2 is given by

Definition 2.2 ([23]). X ⊂ Rn is locally Hp-invex set if and only if, for any x,u ∈ X, there exist a maximum positive number a(x, u) ≤ 1 and a vector function Hp : X × X × [0,1] → Rn, such that

and Hp(x, u; λ) is continuous on the interval (0, a(x, u)), where the logarithm and the exponentials appearing in the relation are understood to be taken componentwise.

Definition 2.3 ([23]). A function f : X → R defined on a locally Hp-invex set X ⊂ Rn is said to be locally (Hp, r)-pre-invex on X if, for any x, u ∈ X, there exists a maximum positive number a(x, u) ≤ 1 such that

where the logarithm and the exponentials appearing in the left-hand side of the inequality are understood to be taken componentwise. If u is fixed, then f is said to be (Hp, r)-pre-invex at u. Correspondingly, if the direction of above inequality is changed to the opposite one, then f is said to (Hp, r)-pre-incave on S or at u.

For convenience, we assume that X be a Hp-invex set, Hp is right differentiable at 0 with respect to the variable λ for each given pair x, u ∈ X, and f : X → R is differential on X. The symbol denotes the right derivative of Hp at 0 with respect to the variable λ for each given pair x, u ∈ X; ∇f(x) ≜ (∇1f(x), ...,∇nf(x))T denotes the differential of f at x, and so

Definition 2.4 ([10]). Let X be a Hp-invex set, Hp is right differentiable at 0 with respect to the variable λ for each given pair x, u ∈ X, and f : X → R is differentiable on X. If for all x ∈ X, one of the relations

hold, then f is said to be (Hp, r)-invex (strictly (Hp, r)-invex) at u ∈ X. If the above inequalities are satisfied at any point u ∈ X then f is said to be (Hp, r)-invex (strictly (Hp, r)-invex) on X.

We now consider the following multiobjective fractional programming problems:

(FP) Minimize

subject to

where f, g : X → Rk and h : X → Rm, f = (f1, f2, ..., fk), g = (g1, g2, ..., gk), h = (h1, h2, ..., hm), are differentiable functions on a (nonempty) Hp-invex set X. Without loss of generality, we can assume that fi(x) ≥ 0, gi(x) > 0, i = 1, 2, ..., k for all x ∈ X. Let X0 = {x ∈ X : h(x) ≤ 0} be the set of all feasible solutions to (FP).

We denote and ϕ(x) = (ϕ1(x), ϕ2(x), ..., ϕk(x)).

Definition 2.5. A feasible solution x∗ ∈ X0 of (FP) is said to be an efficient solution of (FP) if there exist no other feasible solution x ∈ X0 such that

and

It is well known (see, for example [17]) that, if x∗ ∈ X0 is an efficient solution of a multiobjective fractional programming problem (FP), then the following necessary optimality conditions are satisfied:

Theorem 2.1 (Necessary optimality conditions). Let x∗ ∈ X0 be an efficient solution to a multiobjective fractional programming problem (FP) and h satisfies the constraints qualification [17] at x∗. Then, there exist such that

where

The above conditions will be needed in the present analysis.

Remark 2.1 All the theorems in the subsequent parts of this paper will be proved only in the the case when r ≠ 0. The proofs in other cases are easier than in this one, since the differences arise only the form of inequality. Moreover, without loss of generality, we shall assume that r > 0 (in the case when r < 0, the direction some of the inequalities in the proof of the theorems should be changed to the opposite one).

 

3. Parametric duality

We consider the following parametric dual of (FP) as follows:

(DI) Maximize v = (v1, v2, ..., vk)

subject to

Theorem 3.1 (Weak duality). Let x ∈ X0 be a feasible solution for (FP),and let (u, y, z, v) be a feasible solution for (DI). Moreover, we assume that any one of the following conditions holds:

Then

Proof. If the condition (a) holds, then (Hp, r)-invexity of S(.) at u, we have

Using the fundamental property of exponential functions, the above inequality together with (7), imply that

Now suppose contrary to the result that Then

That is,

The above inequalities along with (10) give

By the feasibility of x and from (9) and (10), we have

On adding (12) and (13), we obtain

i.e.,

which contradicts (11).

If condition (b) holds, the from the (Hp, r)-invexity of Q(.) at u,

equivalently

From (13) and (14), we get

The above inequality together with (7) yields

From the (Hp, r)-invexity of P(.) at u, we have

The inequalities (15) and (16), and the fundamental property of exponential functions imply that

That is,

Again if then we get (12) in the same way. But (12) contradicts (17). Therefore, This completes the proof.

Theorem 3.2 (Strong duality). Let x∗ be an efficient solution for (FP) and let h satisfy the constraints qualification [17] at x∗. Then there exist y∗ ∈ Ω, z∗ ∈ Rm and v∗ ∈ Rk such that (x∗, y∗, z∗, v∗) is feasible for (DI).

Also, if the weak duality theorem 3.1 holds for all feasible solutions of the problems (FP) and (DI), then (x∗, y∗, z∗, v∗) is an efficient solution for (DI) and the two objectives are equal at these points.

Proof. Since x∗ is an efficient solution for (FP) and h satisfy the constraints qualification at x∗, there exist y∗ ∈ Ω, z∗ ∈ Rm and v∗ ∈ Rk such that (x∗, y∗, z∗, v∗) satisfies (2)-(6). This, in turn, imply that (x∗, y∗, z∗, v∗) is feasible for (DI). From the weak duality theorem, for any feasible points (x, y, z, v) to (DI), the inequality holds. Hence we conclude that (x∗, y∗, z∗, v∗) is an efficient solution to (DI) and the objective functions of (FP) and (DI) are equal at these points. This completes the proof.

Theorem 3.3 (Strict converse duality). Assume that x∗ and (u∗, y∗, z∗, v∗) be an efficient solution for (FP) and (DI), respectively with Assume that

is strictly (Hp, r)-invex at u∗. Then x∗ = u∗; that is, u∗is an efficient solution for (FP).

Proof. Suppose on the contrary that x∗ ≠ u∗. From (8), (9) and (10), we get

From the strictly (Hp, r)-invexity of A(.), we have

Using the fundamental property of exponential functions, the above inequality together with (7), imply that

Since

i.e.,

By the feasibility of x∗ and (10), we have

Therefore, from (10), (20) and (21), we conclude that

Hence from (19) and (22), we have A(u∗) < 0 which contradicts (18). Hence x∗ = u∗. This completes the proof.

Remark 3.1 The function A(.) in Theorem 3.3 is expressed by the sum of the modified objective part of (FP) and its constraint part If B(.) is strictly (Hp, r)-invex and C(.) is (Hp, r)-invex then the Theorem 3.3 is still holds.

 

4. Parameter free duality

In this section, we take the following form of theorem 2.1:

Theorem 4.1 Let x∗ be an efficient solution to (FP). Assume that h satisfies the constraints qualification at x∗. Then there exist such that

Now we consider the following parameter free dual problem to (FP):

(DII) Maximize

subject to

Denote and Throughout this section, we assume and gi(u) > 0, for all i = 1, 2, ..., k.

Theorem 4.2 (Weak duality). Let x ∈ X0 be a feasible solution for (FP) and let (u, y, z) be a feasible solution for (DII). Assume that

is (Hp, r)-invex at u . Then

Proof. From (Hp, r)-invexity of Θ(.) at u, we have

Using the fundamental property of exponential functions, the above inequality together with (27), imply that

i.e,

Suppose contrary to the result that Then

It follows that

equivalently,

From the feasibility of x, gi(u) > 0 and (28), we have

Therefore (30), implies

i.e,

which contradicts (29). This completes the proof.

Theorem 4.3 (Strong duality). Let x∗ be an efficient solution for (FP) and let h satisfy the constraints qualification [17] at x∗. Then there exist y∗ ∈ Ω and z∗ ∈ Rm such that (x∗, y∗, z∗) is feasible to (DII).

Also, If the weak duality theorem 5.2 holds for all feasible solutions of the problem (FP) and (DII), then (x∗, y∗, z∗) is an efficient solution for (DII) and the two objectives are equal at these points.

Proof. Since x∗ is an efficient solution for (FP) and h satisfy the constraints qualification at x∗, there exist y∗ ∈ Ω and z∗ ∈ Rm such that (x∗, y∗, z∗) satisfies (23)-(26). This, in turn, imply that (x∗, y∗, z∗) is feasible for (DII). From the weak duality theorem 4.2, for any feasible points (x, y, z) to (DII)), the inequality holds. Hence we conclude that (x∗, y∗, z∗) is an efficient solution to (DII) and the objective functions of (FP) and (DII) are equal at these points. This completes the proof.

Theorem 4.4 (Strict converse duality). Assume that x∗ and (u∗, y∗, z∗) be an efficient solution for (FP) and (DII), respectively. Assume that

is strictly (Hp, r)-invex at u∗ Then x∗ = u∗; that is, u∗ is an efficient solution for (FP).

Proof. Suppose on the contrary that x∗ ≠ u∗. From Theorem 4.3, we know that there exist and such that is an efficient solution for (DII) and

By (24), (26) and (31), we obtain

Hence

From (28) and (33), we have

By the feasibility of x∗, gi(u∗) > 0, from (28) and the above inequality, we have

Therefore,

That is,

On the other hand, from strictly (Hp, r)-invexity of U(.) at u∗, we have

The above inequality together with (27) and the fundamental property of the exponential functions yields

which contradicts inequality (34). Hence x∗ = u∗; that is, u∗ is an efficient solution for (FP). This completes the proof.

 

5. Mond-Weir duality

In this section, we consider the following Mond-Weir dual to (FP):

(DIII) Maximize

subject to

Denote and Փ(u) = (Փ1(u),Փ2(u), ...,Փk(u)).

Now we shall state weak, strong and strict converse duality theorems without proof as they can be proved in light of the Theorem 4.2, Theorem 4.3 and Theorem 4.4, proved in previous section.

Theorem 5.1 (Weak duality). Let x ∈ X0 be a feasible solution for (FP) and let (u, y, z) be a feasible solution for (DIII). Assume that

is (Hp, r)-invex at u . Then

Theorem 5.2 (Strong duality). Let x∗ be an efficient solution for (FP) and let h satisfy the constraints qualification [17] at x∗. Then there exist y∗ ∈ I and z∗ ∈ Rm such that (x∗, y∗, z∗) is feasible to (DIII).

Also, If the weak duality theorem 5.1 holds for all feasible solutions of the problem (FP) and (DIII), then (x∗, y∗, z∗) is an efficient solution for (DIII) and the two objectives are equal at these points. Theorem 5.3 (Strict converse duality). Assume that x∗ and (u∗, y∗, z∗) be an efficient solution for (FP) and (DIII), respectively. Assume that

is strictly (Hp, r)-invex at u∗ . Then x∗ = u∗; that is, u∗ is an efficient solution for (FP).

 

6. Conclusion

In this paper, we have used the concept of (Hp, r)-invex functions to established duality results for three type of dual models related to multiobjective fractional programming problem. The question arise whether optimality and duality theorems established in this paper also holds under the assumption of (Hp, r)-invexity for a class of minimax fractional programming problem considered in [1].