OPTIMALITY AND DUALITY IN NONDIFFERENTIABLE MULTIOBJECTIVE FRACTIONAL PROGRAMMING USING α-UNIVEXITY

Abstract

In this paper, a multiobjective nondifferentiable fractional programming problem (MFP) is considered where the objective function contains a term involving the support function of a compact convex set. A vector valued (generalized) ${\alpha}$-univex function is defined to extend the concept of a real valued (generalized) ${\alpha}$-univex function. Using these functions, sufficient optimality criteria are obtained for a feasible solution of (MFP) to be an efficient or weakly efficient solution of (MFP). Duality results are obtained for a Mond-Weir type dual under (generalized) ${\alpha}$-univexity assumptions.

1. Introduction

Most of the real world problems which arise in the areas of portfolio selection, stock cutting, game theory and many decision making problems in management science etc. are (multiobjective) fractional programming problems. Extensive researches have been reported in the literature for the multiobjective nonlinear (nondifferentiable) fractional programming problems involving generalized convex functions by various authors, for details see ([1,4,6-13,15,18-21]) and references therein. The areas which have been explored are mainly to weaken the convexity and to relax the differentiability assumption of the functions used in developing optimality and duality of the above programming problems. Bector et al.[3] introduced univex functions by relaxing the definition of an invex function and obtained optimality and duality results for a nonlinear programming problem. Jayswal[8] defined α-univexity and its generalizations for a real valued function and proved duality theorems for a nondifferentiable generalized fractional programming problem.

Different authors have used different forms of nondifferentiability to obtain optimality conditions and duality theory for fractional programming problem under generalized convexity assumptions. Authors like Mond [15], Singh [19], Zhang and Mond [21] and in the references cited therein considered a class of nondifferentiable fractional programming problems containing square root terms in the objective function and derived optimality criteria and discussed duality theory. Non smooth optimization involves functions for which subderivatives exist [5]. Square root of a positive semidefinite quadratic form is one of the few types of a nondifferentiable function whose subdifferential can be written explicitly. Square root of a quadratic form can be replaced by a more general function, namely, the support function of a compact convex set, whose subdifferential can be simply expressed. For these considerations Mond and Schechter[16] considered programs which contain support function in objective function and studied symmetric duality. Kim et al. [9] established necessary and sufficient optimality conditions and proved duality results for weakly efficient solutions of multiobjective fractional programming problem containing support functions under the assumption of (V, ρ) invex functions. Later in [10], Kim et al. established duality results using (V, ρ) invexity for the same problem with cone constraints.

Motivated by the above researches, in this paper, we consider a nondifferentiable multiobjective fractional programming problem (MFP) over cones with objective function containing support function of a compact convex set. We introduce the concept of α-univexity and its various generalizations for a vector valued function. This generalizes the concept of α-univexity for a scalar valued function[8]. We also give the examples to show the existence of above defined classes of functions. Sufficient optimality conditions for a (weakly) efficient solution of (MFP) are derived using these newly defined classes of (generalized) α-univex functions. A Mond-Weir type dual is proposed for (MFP) and standard duality theorems are proved assuming the functions to be (generalized) α-univex.

 

2. Preliminaries

Let Rn be the n-dimensional Euclidean space and let be its non negative orthant. The following convention for inequalities will be used in this paper. If x, u ∈ Rn, then

Note : For x, u ∈ R, we use x ≤ u to denote x is less than or equal to u.

Definition 2.1 ([17]). A non empty set C in Rn is said to be a cone if for each x ∈ C and λ ≥ 0, λx ∈ C. If in addition C is convex then C is called a convex cone.

Definition 2.2 ([17]). Let C ⊆ Rn be a cone. The set

is called the polar cone of C.

We now consider the following nondifferentiable multiobjective fractional programming problem:

subject to

where ƒ : X → Rk, g : X → Rk and h : X → Rm are continuously differentiable functions over an open subset X of Rn. C1 ⊆ X and C2 are closed convex cones with non empty interiors in Rn and Rm respectively, Di(i = 1, 2, . . . , k) are compact convex sets in Rn and s(x|Di) = max{< x, y > |y ∈ Di} denotes the support function of Di. Let be the set of all feasible solutions of (MFP) and

For any w = (w1,w2, . . . ,wk) ∈ Rn × Rn × . . . × Rn and x ∈ Rn, xTw = (xTw1, xTw2, . . . , xTwk).

We now review some known facts about support functions. The support function s(x|C) of compact convex set C ⊆ Rn, being convex and everywhere finite, has a subgradient at every x in the sense of Rockafellar[17], that is, there exists z ∈ C such that

Equivalently,

The subdifferential of s(x|C) is given by

For any set D ⊆ Rn, the normal cone to D at any point x ∈ D is defined by

If C is a compact convex set then y ∈ NC(x) iff

or equivalently x ∈ ∂s(y|C).

Definition 2.3. A feasible solution is said to be a weakly efficient solution of (MFP) if there does not exist any x ∈ X0 such that

Definition 2.4. A feasible solution is said to be an efficient solution of (MFP) if there does not exist any x ∈ X0 such that

We recall the definition of α-univexity for a differentiable real valued function ƒ : X → R.

Definition 2.5 ([8]). The function f is said to be α-univex with respect to α : X × X → R+ \ {0}, b : X × X → R+, ϕ : R → R and η : X × X → Rn if for every x ∈ X, we have

Now to extend the above concept of α-univexity to multiobjective programming we give the following definitions for a vector valued differentiable function ƒ : X → Rk. Assume that and η : X × X → Rn.

Definition 2.6. The function ƒ : X → Rk is said to be α-univex at with respect to α, b, ϕ, and η if for every x ∈ X and for each i = 1, 2, . . . , k, we have

If (2.1) is a strict inequality for all then ƒ is said to be strict α-univex function.

Definition 2.7. The function ƒ : X → Rk is said to be pseudo α-univex at with respect to α, b, ϕ, and η if for every x ∈ X and for each i = 1, 2, . . . , k, we have

Definition 2.8. The function ƒ : X → Rk is said to be strict pseudo α-univex at with respect to α, b, ϕ, and η if for every and for each i = 1, 2, . . . , k, we have

Definition 2.9. The function ƒ : X → Rk is said to be quasi α-univex at with respect to α, b, ϕ, and η if for every x ∈ X and for each i = 1, 2, . . . , k, we have

ƒ is said to be (strict) α-univex, (strict) pseudo α-univex and quasi α-univex on X if it is (strict) α-univex, (strict) pseudo α-univex and quasi α-univex respectively at every x ∈ X.

We now give the following example to show the existence of vector valued α-univex function.

Example 2.10. Let X = ]0, 1[ and ƒ : X → R2 is given by

Also let, and

Then ƒ is α-univex on X with respect to α, b, ϕ and η.

We note that every α-univex function is pseudo α-univex as well as quasi α-univex but the converse is not true. To illustrate this fact, we give the following examples of pseudo α-univex and quasi α-univex functions which are not α-univex.

Example 2.11. Let and ƒ : X → R2 is given by

Also let, ϕ(x) = 2x, η(x, u) = u − x, α(x, u) = x2 + u

Then ƒ is pseudo α-univex on X with respect to α, b, ϕ and η but it is not α-univex on X because for

Example 2.12. Let and ƒ : X → R2 is given by

Also let, ϕ(x) = 2x, α(x, u) = x + u,

Then ƒ is quasi α-univex on X with respect to α, b, ϕ and η but it is not α-univex on X because for

Now, we give the following lemma.

Lemma 2.13. Assume that ƒ and g are differentiable functions defined from X to Rk, where X is an open subset of Rn and g(x) > 0 for all x ∈ X. If for w = (w1,w2, . . . ,wk) ∈ Rn×Rn×. . .×Rn, ƒ(·)+(·)Tw and −g(·) are α-univex at with respect to α, b, ϕ and η and ϕ is linear, then is α-univex at and η,where

Proof. Consider for each i = 1, 2, . . . , k and x ∈ X,

As ϕ is linear, we have that

Since ƒ(·)+(·)Tw and −g(·) are α-univex at with respect to α, b, ϕ and η, we have for each i = 1, 2, . . . , k,

Therefore,

where Therefore, is α-univex at

Remark 2.1. The following are satisfied:

We now give an example which illustrates the above Lemma 2.13 and Remark 2.14(2).

Example 2.14. Let X =] − 1, 1[ and ƒ : X → R2, g : X → R2 are defined by

Also let, Then is α-univex and hence pseudo α-univex at with respect to as ƒ(·) + (·)Tw and −g(·) are α-univex at with respect to α, b, ϕ and η, where

 

3. Optimality Conditions

The following lemma giving necessary optimality conditions will be used in the sequel. The lemma is cited in [10] and can be obtained from [2] and [9].

Lemma 3.1. (Necessary Optimality Conditions) Let be a weakly efficient solution of (MFP) at which a suitable constraint qualification [14] be satisfied, then there exist and such that

We now establish some sufficient optimality conditions for to be a (weakly) efficient solution of (MFP) under (generalized) α-univexity defined in the previous section.

Theorem 3.2. Let be a feasible solution of (MFP) and that there exist such that

Further assume that all the conditions of Lemma 2.13 are satisfied at and a < 0 ⇒ ϕ(a) < 0. Also if any one of the following conditions hold:

then is a weakly efficient solution of (MFP).

Proof. Suppose that is not a weakly efficient solution of (MFP). Then there exists some x ∈ X0 such that

Using the fact that we have for each i = 1, 2, . . . , k that

Now assume that (a) holds. By Lemma 2.13, is α-univex at with respect to where Since a < 0 ⇒ ϕ(a) < 0 and (3.7) gives

Using the definition of α-univexity of , (3.8) implies

Since therefore multiplying each of the above inequalities by and summing over i = 1, 2, . . . , k, we get that

As (3.4) holds for all x ∈ Rn, we have

Now using (3.10) in (3.9) we get that

Since is α-univex at , the above inequality implies

Using (3.5), we get

As and ϕ0(a) > 0 ⇒ a > 0, we have

But as x is feasible for (MFP) and we get that

which contradicts (3.11). Hence is a weakly efficient solution of (MFP).

Assume that (b) holds. Using Remark 2.14(2), we have that is pseudo α-univex at with respect to where Also as a < 0 ⇒ ϕ(a) < 0, (3.7) gives

Now being pseudo α-univex at , (3.12) implies

Since therefore multiplying each of the above inequalities by and summing over i = 1, 2, . . . , k, we get

As (3.4) holds for all x ∈ Rn, we have

Because x is feasible for (MFP) and we get that

Using (b), (3.5) and above inequality, we obtain

As is quasi α-univex at , we obtain from above inequality that

Adding (3.13) and (3.15), we get that

which contradicts (3.14). Hence is a weakly efficient solution of (MFP).

Theorem 3.3. Let be a feasible solution of (MFP) and that there exist such that the conditions (3.4) - (3.6) hold. Assume that all the conditions of Lemma 2.13 are satisfied at for all i = 1, 2, . . . , k and a < 0 ⇒ ϕ(a) < 0. Also assume that is α-univex at with respect to α, b0, ϕ0 and η with and ϕ0(a) > 0 ⇒ a > 0. Then is an efficient solution of (MFP).

Proof. Suppose that is not an efficient solution of (MFP). Then there exists some x ∈ X0 such that

Using (3.6) and the fact that we have for all i = 1, 2, . . . , k, i ≠ j that

and

That is,

Since and −g(·) are α-univex at with respect to α, b, ϕ and η, therefore by Lemma 2.13, is α-univex at with respect to where for all i = 1, 2, . . . , k. Using the assumption that a < 0 ⇒ ϕ(a) < 0 where ϕ is linear, (3.16) and (3.17) give

Using α-univexity of in last two inequalities, we get

Since therefore multiplying each of the above inequalities by and summing over i = 1, 2, . . . , k, we get

Rest of the proof follows on the lines of proof of part (a) of Theorem 3.2.

Theorem 3.4. Let be a feasible solution of (MFP) and that there exist Let such that the conditions (3.4) - (3.6) hold. Assume that all the conditions of Lemma 2.13 are satisfied at being strict α-univex at . Further assume that is quasi α-univex at with respect to α, b0, ϕ0 and η. Also let a ≤ 0 ⇒ ϕ(a) ≤ 0 and a ≤ 0 ⇒ ϕ0(a) ≤ 0. Then x is an efficient solution of (MFP).

Proof. Suppose that is not an efficient solution of (MFP). Then there exists some x ∈ X0 such that

Using (3.6) and the fact that we have for all i = 1, 2, . . . , k, i ≠ j that

and

That is,

Since is strict α-univex and −g(·) is α-univex at with respect to α, b, ϕ and η, therefore by Remark 2.14(3), is strict pseudo α-univex at with respect to where for all i = 1, 2, . . . , k. Thus by using the assumption that a ≤ 0 ⇒ ϕ(a) ≤ 0, (3.18) and (3.19) give

Using the definition of strict pseudo α-univexity of , (3.20) implies

Since therefore multiplying each of the above inequalities by and summing over i = 1, 2, . . . , k, we get

Rest of the proof follows on the lines of proof of part (b) of Theorem 3.2.

 

4. Duality

Now we consider the following Mond-Weir type dual of (MFP).

(MFD) Maximize

subject to

where wi (i = 1, 2, . . . , k) is a vector in Rn and uTw = (uTw1, . . . , uTwk).

Theorem 4.1. (Weak Duality) Let x be feasible for (MFP) and (u, y, λ,w) be feasible for (MFD). Assume that

then

Proof. Let us suppose on the contrary that

that is,

Using the fact that s(x|Di) ≥ xTwi for all i = 1, 2, . . . , k, we get that

By (a) as and −g(·) are α-univex at u with respect to α, b, ϕ and η, therefore by Lemma 2.13, is α-univex at u with respect to where for all i = 1, 2, . . . , k. Thus by using the assumption that a < 0 ⇒ ϕ(a) < 0, (4.4) gives

Now as is α - univex at u, we get

Since λ ≥ 0 by (4.3), therefore multiplying above inequality for each i = 1, 2, . . . , k by λi and summing over i = 1, 2, . . . , k, we get that

From the dual constraint (4.1), we have

therefore there exist such that

On using (4.6) in (4.2), we obtain

Since x is feasible for (MFP), y ∈ C2 and we have

therefore (4.7) and (4.8) together give

From assumption (b) as a ≤ 0 ⇒ ϕ0(a) ≤ 0 and b0(x, u) ≥ 0, therefore

Since yT h(·) + vT (·) is α-univex at u with respect to α, b0, ϕ0 and η, therefore the above inequality gives

which on using (4.6) reduces to

This contradicts (4.5). Hence,

Theorem 4.2. (Strong Duality) Let be a weakly efficient solution of (MFP) at which a suitable constraint qualification [14] be satisfied. Then there exist such that is feasible for (MFD) and the objective function values of (MFP) and (MFD) are equal. Furthermore, if the assumptions of weak duality Theorem 4.1 hold for all the feasible solutions of (MFP) and (MFD), then is weakly efficient for (MFD).

Proof. Since is a weakly efficient solution of (MFP), therefore there exist such that (3.1),(3.2) and (3.3) hold. Since ∈ C1 and C1 is a closed convex cone, therefore for any x ∈ C1, we have Thus replacing x by in (3.1), we get

That is,

Also by letting x = 0 and in (3.1), we get

Therefore by using (3.2) and (4.10), we have that

Thus (4.9) and (4.11) imply that is feasible for (MFD) and the objective function values of (MFP) and (MFD) are equal. Since the assumptions of weak duality hold for all the feasible solutions of (MFP) and (MFD), we get that is a weakly efficient solution of (MFD).

 

5. Conclusion

This paper generalizes the concept of α-univexity for a real valued function by defining the concept of α-univexity, pseudo α-univexity and quasi α-univexity for a vector valued function. Examples have been included to show the existence of these functions. Sufficient optimality criteria have been obtained for (MFP) by using the above defined classes of (generalized) α-univex functions. Assuming the functions to be α-univex duality is established between (MFP) and its Mond-Weir type dual (MFD).