VECTOR OPTIMIZATION INVOLVING GENERALIZED SEMILOCALLY PRE-INVEX FUNCTIONS

Abstract

In this paper, a vector optimization problem over cones is considered, where the functions involved are $\eta$-semidifferentiable. Necessary and sufficient optimality conditions are obtained. A dual is formulated and duality results are proved using the concepts of cone $\rho$-semilocally preinvex, cone $\rho$-semilocally quasi-preinvex and cone $\rho$-semilocally pseudo-preinvex functions.

1. Introduction

Ewing [1] introduced the concept of semilocally convex functions. It was further extended to semilocally quasiconvex, semilocally pseudoconvex functions by Kaul and Kaur [2]. Necessary and sufficient optimality conditions were derived by Kaul and Kaur [3,4], and Suneja and Gupta [8].

Weir and Mond [13] considered preinvex functions for multiple objective optimization. Further Weir and Jeyakumar [12] introduced the class of cone-preinvex functions and obtained optimality conditions and duality theorems for a scalar and vector valued programs. Weir [11] introduced cone-semilocally convex functions and studied optimality and duality theorems for vector optimization problems over cones. Preda and Stancu-Minasian [5,6,7] studied optimality and duality results for a fractional programming problem where the functions involved were semilocally preinvex.

In the recent years Suneja et al. [9] introduced the concepts of ρ-semilocally preinvex and related functions and obtained optimality and duality for multiobjective non-linear programming problem, Suneja and Bhatia [10] defined cone-semilocally preinvex and related functions. They obtained necessary and sufficient optimality conditions for a vector optimization problem over cones. In this paper, we have defined cone ρ-semilocally preinvex, cone ρ-semilocally quasipreinvex, cone ρ-semilocally pseudopreinvex functions and established necessary and sufficient optimality conditions for a vector optimization problem over cones.

 

2. Definitions and Preliminaries

Let S ⊆ Rn and η : S × S → Rn and θ : S × S → Rn be two vector valued functions.

Definition 2.1. The set S ⊆ Rn is said to be η-locally star shaped set at x∗ ∈ S if for each x ∈ S there exists a positive number a η (x, x∗) ≤ 1 such that x∗ + λη(x,x∗) ∈ S, for 0 ≤ λ ≤ aη (x,x∗).

Definition 2.2 ([10]). Let S ⊆ Rn be an η-locally star shaped set at x∗ ∈ S and K ⊆ Rm be a closed convex cone with non-empty interior. A vector valued function f : S → Rm is said to be K-semilocally preinvex (K-Slpi) at x∗ with respect to η if corresponding to x∗ and each x ∈S, there exist a positive number dη (x,x∗ ) ≤ aη (x, x∗ ) ≤ 1 such that

We now introduce ρ semilocally preinvex functions over cones.

Definition 2.3. Let S ⊆ Rn be an η-locally star shaped set at x∗ ∈ S,ρ ∈ Rmand K ⊆ Rm be a closed convex cone with nonempty interior. A vector valued function f : S → Rm is said to be ρ-semilocally preinvex over K(kρ-Slpi) at x∗ ∈ S with respect to η if corresponding to x∗ and each x ∈ S, there exists a positive number dη (x,x ∗ ) ≤ aη (x,x∗ ) ≤ 1 such that

Remark 2.1. If ρ = 0 the definition of Kρ-Slpi function reduces to that of K-slpi function given by Suneja and Meetu [10].

If K = R+ , the definition of Kρ-slpi function reduces to that of ρ-slpi function given by Suneja et al. [9]. In addition if η(x,x∗ ) = x − x∗ then Kρ-semilocally preinvex functions reduces to K-semilocally convex functions defined by Weir [11].

We now give an example of a function which is Kρ-slpi but fails to be ρ-slpi.

Example 2.1. We consider the following η-locally star shaped set as given by Suneja and Meetu [10]. Let S = R ╲ E, where

θ(x,x∗ ) = x − x∗

Consider the function f : S → R2 defined by

Then f is Kρ-slpi at x∗ = −1. But f is not ρ-slpi because for x = 1,

Definition 2.4. The function f : S → Rm is said to be η-semidifferentiable at x∗ ∈ S if

exists for each x ∈ S.

Theorem 2.1. If f is Kρ-Slpi at x∗ then

Proof. Since the function f is Kρ-slpi at x∗ with respect to η therefore corresponding to each x ∈ S there exists a positive number

such that

which implies

Since K is a closed cone, therefore by taking limit as λ → 0+ , we get

We now introduce Kρ-semilocally naturally quasi preinvex (Kρ-slnqpi) over cones.

Definition 2.5. The function f is said to be Kρ-semilocally naturally quasi preinvex (Kρ-Slnqpi) at x∗ with respect to η if

Theorem 2.2. If f is Kρ-slpi at x∗ ∈ S with respect to η then f is Kρ-slnqpi at x∗ with respect to same η.

Proof. Let f be Kρ-slpi at x∗ , then there exists a positive number d η (x,x∗) ≤ aη (x,x∗) such that

Suppose that

then

Adding (2.1) and (2.2) we get

Since K is a closed cone, therefore taking limit as λ → 0+, we get

Thus

But the converse is not true as shown in the following example.

Example 2.2. Consider set S = R / E, where E = Then as discussed in Example 2.1, S is η-locally star shaped. Consider the function f : S → R2 defined by

θ(x, x∗ ) = x − x∗ .

Then function f is Kρ-slnqpi at x∗ = −2, for ρ = (1, 0), where

because

But the function f fails to bekρ-slpi at x∗ = −2 by Theorem 2.1 because for x = 1,

Definition 2.6. The functionf : S → Rm is said to be Kρ-semilocally quasi preinvex (Kρ-slqpi) at x∗ with respect to η if

Remark 2.2. The following diagram illustrates the relation among Kρ-slpi function, Kρ-slnqpi and Kρ-slqpi functions.

Figure 1

We now give an example of a function which is Kρ-slnqpi but fails to be kρ-slqpi.

Example 2.3. The function f considered in Example 2.2 is Kρ-slnqpi at x∗ = −2. But fails to be Kρ-slqpi at x∗ = −2 because for x = 1

but

The next definition introduces cone semilocally pseudo preinvex functions over cone.

Definition 2.7. The function f : S → Rm is said to be Kρ-semilocally pseudo preinvex (Kρ-slppi) at x∗ , with respect to η if

 

3. Optimality Conditions

Consider the following Vector Optimization Problem

where f : S → Rm and g : S → Rp are η-semidifferentiable functions with respect to same η and S ⊆ Rn is a nonempty η-locally star shaped set.

Let K ⊆ Rm and Q ⊆ Rp be closed convex cones having non-empty interior and let X = {x ∈ S : −g(x) ∈ Q} be the set of all feasible solutions of (VOP).

Definition 3.1. A point x∗ ∈ X is called

We will use the following Alternative Theorem given by Weir and Jeyakumar [12].

Theorem 3.1. Let X, Y be real normed linear spaces and K be a closed convex cone in Y with nonempty interior, let S ⊆ X. Suppose that f : S → Y be K-preinvex. Then exactly one of the following holds:

where int denotes interior and K∗ is the dual cone of K.

We now establish the necessary optimality conditions for (VOP).

Theorem 3.2 (Fritz John Type Necessary Optimality Conditions). Let x∗ ∈ X be a weak minimum of (VOP) and suppose (df) + (x∗ , η(x,x∗)) and (dg) + (x∗ ,η(x,x∗)) are K-preinvex and Q-preinvex functions of x respectively with respect to same η(x,x∗) and η(x∗ , x∗) = 0 then there exists τ* τ∗ ∈ K∗ , µ∗ ∈ Q∗ such that

Proof. We assert that the system

has no solution x ∈ S, where

F(x) = ((df) + (x∗ , η(x,x∗)), (dg) + (x∗ , η(x,x∗)) + g(x∗)).

If possible, let there be a solution x0 ∈ S of (3.3). Then

− F(x0) ∈ int(K × Q) ⇒ −(df) + (x∗ ,η(x0,x∗ )) ∈ int K

and

−(dg) + (x∗ ,η(x0,x∗ )) − g(x∗) ∈ int Q.

Since S is locally star shaped and x∗ ,x0 ∈ S, therefore we can find λ 0 > 0 such that for λ ∈ (0,λ0),

x∗ + λη(x0 ,x∗ ) ∈ S.

By definition of (df) + (x∗ , η(x,x∗)) and (dg) + (x∗,η(x,x∗)), it follows that

−[f(x∗ + λη(x0 ,x∗ )) − f(x∗ )] ∈ int K

and

− [g(x∗ + λη(x0,x∗)) − g(x∗)] − g(x∗ ) ∈ int Q.

⇒ f(x∗) − f(x∗ + λη(x0,x∗ )) ∈ int K

and

−g(x∗ + λη(x0 ,x∗)) ∈ int Q, for λ ∈ (0, λ0),

which is a contradiction as x∗ is a weak minimum of (VOP). Hence the system (3.3) has no solution x ∈ S.

Also F is (K ×Q) preinvex on S as (df) + (x∗,η(x,x∗)) and (dg) + (x∗,η(x,x∗)) are K-preinvex and Q-preinvex on S respectively. Therefore, by Theorem 3.1, there exists τ∗ ∈ K∗ and µ∗ ∈ Q∗ not both zero such that

Taking x = x∗ , we get

Also µ∗ ∈ Q∗ and −g(x∗) ∈ Q, implies that

From (3.5) and (3.6), we get

µ∗Tg(x∗) = 0.

From (3.4), we get

τ∗T(df) + (x∗,η(x,x∗)) + µ∗T(dg) + (x∗,η(x,x∗)) ≥ 0, for all x ∈ S. ὆

We use the following Slater type constraint qualification to prove the KuhnTucker type necessary optimality conditions for (VOP).

Definition 3.2. The function g is said to satisfy Slater type constraint qualification at x∗if g is Q-preinvex at x∗ and there exists ∈Ssuch that−g()∈ intQ.

Theorem 3.3 (Kuhn Tucker Type Necessary Optimality Conditions). Let x∗ ∈ X be a weak minimum of (VOP) and suppose (df) + (x∗,η(x,x∗)) and (dg) + (x∗,η(x,x∗))are K-preinvex and Q-preinvex functions of x respectively with respect to the same η(x,x∗). Suppose that g is Q-slpi at x∗ and g satisfies Slater type constraint qualification at x∗ and η(x∗,x∗) = 0, then there exists 0 ≠ τ∗ ∈ K∗,µ∗ ∈ Q∗ such that (3.1) and (3.2) hold.

Proof. Since x∗ is a weak minimum of (VOP), therefore by Theorem 3.2, there exist τ∗ ∈ K∗ , µ∗ ∈ Q∗ such that (3.1) and (3.2) hold. If possible, let τ∗ = 0, then from (3.1), we get

Since g is Q-slpi at x∗ , therefore we have

Adding (3.7) and (3.8) and using (3.2), we get

Again by Slater type constraint qualification, there exists ∈ S such that

which is a contradiction to (3.9). Hence τ∗≠ 0. ΢

Now we will establish some sufficient conditions for (VOP).

Theorem 3.4. If x∗ ∈ X, f is Kρ-slpi and g is Qσ-slpi at x∗ and there exist 0 ≠ τ∗ ∈ K∗ and µ∗ ∈ Q∗ satisfying the conditions (3.1) and (3.2),then x∗ is a weak minimum of (VOP) provided

τ∗T ρ + µ∗T σ ≥ 0.

Proof. Suppose that x∗ is not a weak minimum of (VOP), then there exists x ∈ X such that

f (x∗ ) − f (x) ∈ int K .

Since 0 ≠ τ∗ ∈ K∗ , it follows that

Since f is Kρ-slpi and g is Qσ-slpi at x∗ , therefore

and

which contradicts (3.10). ΢

Theorem 3.5. Let x ∈ X. If there exist 0 ≠ τ∗ ∈ K∗ , µ∗ ∈ Q∗ satisfying the conditions (3.1) and (3.2), g is Qσ-slqpi at x∗ and f is Kρ-slppi at x∗ then x∗ is a weak minimum of (VOP) provided

τ∗T ρ + µ∗T σ ≥ 0 .

Proof. Let x ∈ X and suppose µ∗ ≠ 0. Then −g(x) ∈ Q implies that

µ∗Tg(x) ≤ 0.

From condition (3.2), it follows that

µ∗T (g(x) − g(x∗ )) ≤ 0,

which gives that

g(x) − g(x∗ ) ∉ int Q.

Also g is Qσ-slqpi at x∗ , therefore, we get

If µ∗ = 0, then the above inequality holds trivially.

On using (3.1), we have

Since f is Kρ-slppi at x∗ , we get

− (f(x) − f(x∗)) ∉ int K ⇒ f(x∗ ) − f (x) ∉ int K.

Thus x∗ is a weak minimum of (VOP). ΢

 

4. Duality

We associate the following Mond-Weir type dual with (VOP),

(VOD) K-maximize f(u)

Theorem 4.1 (Weak Duality). Let x ∈ X and (u, τ, µ) be dual feasible, suppose f is Kρ-slppi and g is Qσ-slqpi at u then

f(u) − f(x) ∉ int K,

provided τ ρ + µσ ≥ 0.

Proof. Since x ∈ X and (u, τ, µ) is dual feasible, therefore, we get

µT (g(x) − g(u)) ≤ 0

If µ ≠ 0, then the above inequality gives

g(x) − g(u) ∉ int Q.

Since g is Qσ-slqpi at u, we get

If µ = 0, then the above inequality holds trivially. Now using (4.1), we get

Since f is Kρ-slppi at u, we get

− (f(x) − f(u)) ∉ int K ⇒ (f(u) − f(x)) ∉ int K.

Thus u is a weak minimum of (VOD). ΢

Theorem 4.2 (Strong Duality). Let x∗ be a weak minimum of (VOP), (df) + (u,η(x,u)) be K-preinvex and (dg) + (u,η(x,u)) be Q-preinvex functions on S. Suppose slater type constraint qualification holds at x∗ . Then there exist 0 ≠ τ∗ ∈ K∗ , µ∗ ∈ Q∗ such that (x∗ , τ∗ , µ∗)is feasible for (VOD). Moreover, if for each feasible (u, τ, µ) of (VOD), hypothesis of above theorem holds then(x∗ , τ∗ , µ∗) is a weak maximum of (VOD).

Proof. Since all the conditions of Theorem 3.3 hold, therefore, there exist 0 ≠ τ∗ ∈ K∗ , µ∗ ∈ Q∗ such that (3.1) and (3.2) hold. This implies that (x∗ , τ∗ , µ∗) is feasible for (VOD). If possible let (x∗ , τ∗ , µ∗) be not a weak maximum of (VOD), then there exists (u, τ, µ) feasible for (VOD) such that

f (u) −f (x∗) ∈ int K.

But this is a contradiction to weak duality result as x∗ ∈ X and (u, τ, µ) is feasible for (VOD). Hence (x∗ , τ∗ , µ∗) must be a weak maximum of (VOD). ΢