On ε-optimality conditions for multiobjective fractional optimization problems

Abstract

A multiobjective fractional optimization problem (MFP), which consists of more than two fractional objective functions with convex numerator functions and convex denominator functions, finitely many convex constraint functions, and a geometric constraint set, is considered. Using parametric approach, we transform the problem (MFP) into the non-fractional multiobjective convex optimization problem (NMCP) _v with parametric v ∈ ℝ ^p , and then give the equivalent relation between (weakly) ε-efficient solution of (MFP) and (weakly) -efficient solution of . Using the equivalent relations, we obtain ε- optimality conditions for (weakly) ε- efficient solution for (MFP). Furthermore, we present examples illustrating the main results of this study.

2000 Mathematics Subject Classification: 90C30, 90C46.

1 Introduction

We need constraint qualifications (for example, the Slater condition) on convex optimization problems to obtain optimality conditions or ε- optimality conditions for the problem.

To get optimality conditions for an efficient solution of a multiobjective optimization problem, we often formulate a corresponding scalar problem. However, it is so difficult that such scalar program satisfies a constraint qualification which we need to derive an optimality condition. Thus, it is very important to investigate an optimality condition for an efficient solution of a multiobjective optimization problem which holds without any constraint qualification.

Jeyakumar et al. [1, 2], Kim et al. [3], and Lee et al. [4], gave optimality conditions for convex (scalar) optimization problems, which hold without any constraint qualification. Very recently, Kim et al. [5] obtained ε- optimality theorems for a convex multiobjective optimization problem. The purpose of this article is to extend the ε- optimality theorems of Kim et al. [5] to a multiobjective fractional optimization problem (MFP).

Recently, many authors [5–15] have paid their attention to investigate properties of (weakly) ε- efficient solutions, ε- optimality conditions, and ε- duality theorems for multiobjective optimization problems, which consist of more than two objective functions and a constrained set.

In this article, an MFP, which consists of more than fractional objective functions with convex numerator functions, and convex denominator functions and finitely many convex constraint functions and a geometric constraint set, is considered. We discuss ε- efficient solutions and weakly ε- efficient solutions for (MFP) and obtain ε- optimality theorems for such solutions of (MFP) under weakened constraint qualifications. Furthermore, we prove ε- optimality theorems for the solutions of (MFP) which hold without any constraint qualifications and are expressed by sequences, and present examples illustrating the main results obtained.

2 Preliminaries

Now, we give some definitions and preliminary results. The definitions can be found in [16–18]. Let g : ℝ ⁿ → ℝ ∪ {+∞} be a convex function. The subdifferential of g at a is given by

where domg: = {x ∈ ℝ ⁿ | g(x) < ∞} and ⟨·, ·⟩ is the scalar product on ℝ ⁿ . Let ε ≧ 0. The ε- subdifferential of g at a ∈ domg is defined by

The conjugate function of g : ℝ ⁿ → ℝ ∪ {+∞} is defined by

The epigraph of g, epig, is defined by

For a nonempty closed convex set C ⊂ ℝ ⁿ , δ _C : ℝ ⁿ → ℝ ∪ {+∞} is called the indicator of C if .

Lemma 2.1[19]If h : ℝ ⁿ → ℝ ∪ {+∞} is a proper lower semicontinuous convex function and if a ∈ domh, then

Lemma 2.2[20]Let h : ℝ ⁿ → ℝ be a continuous convex function and u : ℝ ⁿ → ℝ ∪ {+∞} be a proper lower semicontinuous convex function. Then

Now, we give the following Farkas lemma which was proved in [2, 5], but for the completeness, we prove it as follows:

Lemma 2.3 Let h _i : ℝ ⁿ → ℝ, i = 0, 1, ⋯, l be convex functions. Suppose that {x ∈ ℝ ⁿ | h _i (x) ≦ 0, i = 1, ⋯, l} ≠ ∅. Then the following statements are equivalent:

(i) {x ∈ ℝ ⁿ | h _i (x) ≦ 0, i = 1, ..., l} ⊆ {x ∈ ℝ ⁿ | h₀(x) ≧ 0}

(ii).

Proof. Let Q = {x ∈ ℝ ⁿ | h _i (x) ≦ 0, i = 1, ..., l}. Then Q ≠ ∅ and by Lemma 2.1 in [2], . Hence, by Lemma 2.2, we can verify that (i) if and only if (ii).

Lemma 2.4[16]Let h _i : ℝ ⁿ → ℝ ∪ {+∞}, i =, 1, ⋯, m be proper lower semi-continuous convex functions. Let ε ≧ 0. if, where ri domh _i is the relative interior of domh _i , then for all,

3 ε-optimality theorems

Consider the following MFP:

Let f _i : ℝ ⁿ → ℝ, i = 1, ..., p be convex functions, g _i : ℝ ⁿ → ℝ, i = 1, ..., p, concave functions such that for any x ∈ Q, f _i (x) ≧ 0 and g _i (x) > 0, i = 1, ..., p, and h _j : ℝ ⁿ → ℝ, j = 1, ..., m, convex functions. Let ε = (ε₁, ..., ε _p ), where ε _i ≧ 0, i = 1, ..., p.

Now, we give the definition of ε- efficient solution of (MFP) which can be found in [11].

Definition 3.1 The pointis said to be an ε-efficient solution of (MFP) if there does not exist x ∈ Q such that

When ε = 0, then the ε- efficiency becomes the efficiency for (MFP) (see the definition of efficient solution of a multiobjective optimization problem in [21]).

Now, we give the definition of weakly ε- efficient solution of (MFP) which is weaker than ε- efficient solution of (MFP).

Definition 3.2 A pointis said to be a weakly ε-efficient solution of (MFP) if there does not exist x ∈ Q such that

When ε = 0, then the weak ε- efficiency becomes the weak efficiency for (MFP) (see the definition of efficient solution of a multiobjective optimization problem in [21]).

Using parametric approach, we transform the problem (MFP) into the nonfractional multiobjective convex optimization problem (NMCP) _v with parametric v ∈ ℝ ^p :

Adapting Lemma 4.1 in [22] and modifying Proposition 3.1 in [12], we can obtain the following proposition:

Proposition 3.1 Let. Then the following are equivalent:

(i)is an ε-efficient solution of (MFP).

(ii)is an-efficient solution of , where and .

(iii) or

where.

Proof. (i) ⇔ (ii): It follows from Lemma 4.1 in [22].

• (ii) ⇒ (iii): Let be an -efficient solution of , where and . Then or . Suppose that . Then for any and all i = 1, . . . p,

  

Hence the -efficiency of yields

for any and all i = 1, ..., p. Thus we have, for all ,

• (iii) ⇒ (ii): Suppose that . Then there does not exist x ∈ Q such that ; that is, there does not exist x ∈ Q such that

  

for all i = 1, ..., p. Hence, there does not exist x ∈ Q such that

Therefore, is an -efficient solution of , where .

Assume that . Then, from this assumption

(3.1)

for any . Suppose to the contrary that is not an -efficient solution of . Then, there exist and an index j such that

Therefore, and , which contradicts the above inequality. Hence, is an -efficient solution of .

We can easily obtain the following proposition:

Proposition 3.2 Letand suppose that. Then the following are equivalent:

(i)is a weakly ε-efficient solution of (MFP).

(ii)is a weakly-efficient solution of, whereand.

(iii) there exists such that

Proof. (i) ⇔ (ii): The proof is also following the similar lines of Proposition 3.1.

• (ii) ⇒ (iii): Let φ(x) = (φ ₁(x), ..., φ _p (x)), ∀x ∈ Q, where . Then, φ _i (x), i = 1,⋯, p, are convex. Since is a weakly ε- efficient solution of , where , , and hence, it follows from separation theorem that there exist , i = 1, ..., p, such that

  

Thus (iii) holds.

• (iii) ⇒ (ii): If (ii) does not hold, that is, is not a weakly -efficient solution of , then (iii) does not hold. □

We present a necessary and sufficient ε-optimality theorem for ε-efficient solution of (MFP) under a constraint qualification, which will be called the closedness assumption.

Theorem 3.1 Letand assume thatandi = 1, ..., p. Suppose that

is closed, where, i = 1, ..., p. Then the following are equivalent.

(i)is an ε-efficient solution of (MFP).

(ii)

(iii) there exist α _i ≧ 0, , β _i ≧ 0, , i = 1, ..., p, λ _j ≧ 0, γ _j ≧ 0, , j = 1, ..., m, μ _i ≧ 0, q _i ≧ 0, , z _i ≧ 0, i = 1, ..., p such that

and

Proof. Let .

• (i) ⇔ (by Proposition 3.1) h ₀(x) ≧ 0, .

⇔ , i = 1, ..., p, h _j (x) ≦ 0, j = 1, ..., m} ⊂ {x | h₀(x) ≧ 0}.

⇔ (by lemma 2.3)

Thus by the closedness assumption, (i) is equivalent to (ii).

• (ii) ⇔ (iii): (ii) ⇔ (by Lemma 2.1), there exist α _i ≧ 0, , i = 1, ..., p, β _i ≧ 0, , i = 1, ..., p, λ _j ≧ 0, γ _j ≧ 0, , j = 1, ..., m, μ _i ≧ 0, q _i ≧ 0, , i = 1, ..., p, z _i ≧ 0, , i = 1, ..., p such that

  

⇔ there exist α _i ≧ 0, , β _i ≧ 0, , i = 1, ..., p, λ _j ≧ 0, γ _j ≧ 0, , j = 1, ..., m, μ _i ≧ 0, q _i ≧ 0, , z _i ≧ 0, i = 1, ..., p such that

and

⇔ (iii) holds. □

Now we give a necessary and sufficient ε-optimality theorem for ε-efficient solution of (MFP) which holds without any constraint qualification.

Theorem 3.2 Let. Suppose thatand, i = 1, ..., p. Thenis an ε-efficient solution of (MFP) if and only if there exist α _i ≧ 0, , i = 1, ..., p, β _i ≧ 0, , i = 1, ..., p, , , , j = 1, ..., m, , , , , , k = 1, ..., p such that

and

Proof. is an ε-efficient solution of (MFP)

⇔ (from the proof of Theorem 3.1)

⇔ (by Lemma 2.1) there exist α _i ≧ 0, , i = 1, ..., p, β _i ≧ 0, , i = 1, ..., p, , , , j = 1, ..., m, , , , , , k = 1, ..., p, such that

⇔ there exist α _i ≧ 0, , i = 1, ..., p, β _i ≧ 0, , i = 1, ..., p, , , , j = 1, ..., m, , , , , , k = 1, ..., p, such that

and

We present a necessary and sufficient ε-optimality theorem for weakly ε-efficient solution of (MFP) under a constraint qualification.

Theorem 3.3 Letand assume that, i = 1, ..., p, andis closed. Then the following are equivalent.

(i)is a weakly ε-efficient solution of (MFP).

(ii) there exist μ _i ≧ 0, i = 1, ..., p, such that

where, i = 1, ..., p.

(iii) there exist μ _i ≧ 0, , α _i ≧ 0, , β _i ≧ 0, , i = 1, ..., p, λ _j ≧ 0, γ _j ≧ 0, , j = 1, ..., m, such that

and

Proof. (i) ⇔ (ii): is a weakly ε-efficient solution of (MFP)

⇔ (by Proposition 3.2) there exist μ _i ≧ 0, i = 1, ..., p, such that

⇔ there exist μ _i ≧ 0, i = 1, ..., p, such that

⇔ (by Lemma 2.3) there exist μ _i ≧ 0, i = 1, ..., p, such that

Thus, by the closedness assumption, (i) is equivalent to (ii).

• (ii) ⇔ (iii): (ii) ⇔ (by Lemma 2.1) there exist μ _i ≧ 0, , α _i ≧ 0, , β _i ≧ 0, , i = 1, ..., p, λ _j ≧ 0, γ _j ≧ 0, , j = 1, ..., m, such that

  

⇔ (iii) holds. □

Now, we propose a necessary and sufficient ε-optimality theorem for weakly ε-efficient solution of (MFP) which holds without any constraint qualification.

Theorem 3.4 Letand assume that. Thenis a weakly ε-efficient solution of (MFP) if and only if there exist μ _i ≧ 0, i = 1, ..., p, , α _i ≧ 0, , i = 1, ..., p, β _i ≧ 0, , i = 1, ..., p, , , , j = 1, ..., m, such that

and

Proof. is a weakly ε-efficient solution of (MFP)

⇔ ((from the proof of Theorem 3.3) there exist μ _i ≧ 0, i = 1, ..., p, such that

⇔ (by Lemma 2.1) there exist μ _i ≧ 0, i = 1, ..., p, , α _i ≧ 0, , i = 1, ..., p, β _i ≧ 0, , i = 1, ..., p, , , , j = 1, ..., m, such that

⇔ there exist μ _i ≧ 0, i = 1, ..., p, , α _i ≧ 0, , i = 1, ..., p, β _i ≧ 0, , i = 1, ..., p, , , , j = 1, ..., m, such that

and

□

Now, we give examples illustrating Theorems 3.1, 3.2, 3.3, and 3.4.

Example 3.1 Consider the following MFP:

Let, and f₁(x₁, x₂) = x₁, g₁(x₁, x₂) = 1, f₂(x₁, x₂) = x₂, g₂(x₁, x₂) = x₁, h₁(x₁, x₂) = -x₁+ 1 and h₂(x₁, x₂) = -x₂ + 1.

(1)Let. Thenis an ε-efficient solution of (MFP)₁.

Letand. Then, and

Thus,. It is clear thatand. Let. Then

where coD is the convexhull of a set D and cone coD is the cone generated by coD. Thus A is closed. Let. Then

B = {(1, 0)} × [0, ∞)+{(0, 0)} × [1, ∞)+{(0, 1)} × [0, ∞)+{(-1, 0)} × [0, ∞)+A. Since (0,-1,-1) ∈ A, (0, 0, 0) ∈ B. Thus (ii) of Theorem 3.1 holds. Let α₁ = β₁ = γ₁ = q₁ = z₁ = α₂ = β₂ = γ₂ = q₂ = z₂ = 0, and let μ₁ = μ₂ = 1, and λ₁ = 0 and λ₁ = 2. Moreover, , , , , , , , , .

Thus,and.

Thus, (iii) of Theorem 3.1 holds.

(2) Let. Thenis not an ε-efficient solution of (MFP)₁, butis a weakly ε-efficient solution of (MFP)₁.

Let. Then

Hence, C is closed. Moreover,, and. Letand. Then,, . Let μ₁ = 1 and μ₂ = 1. Then,

Since (-1, 0,-1) ∈ C, . So, (ii) of Theorem 3.3 holds. Let α₁ = β₁ = γ₁ = α₂ = β₂ = γ₂ = 0, λ₁ = 1 and λ₂ = 0. Then,

and

Thus, (iii) of Theorem 3.3 holds.

Example 3.2 Consider the following MFP:

Let, and f₁(x₁, x₂) = -x₁ + 1, g₁(x₁, x₂) = 1, f₂(x₁, x₂) = x₂, g₂(x₁, x₂) = -x₁ + 1, h₁(x₁, x₂) = [max{0, x₁}]²and h₂(x₁, x₂) = -x₂ + 1.

(1) Let. Then,is an ε-efficient solution of (MFP)₂. Let. Then, clA = cone co{(0, -1, -1), (1, 0, 0), (-1, 0, 0), (1, 1, 1), (0, 0, 1)}. Here, (1, 0, 0) ∈ clA, but (1, 0, 0) ∈ A, where clA is the closure of the set A. Thus, A is not closed. Let Q = {(x₁, x₂) ∈ ℝ ⁿ | h₁(x₁, x₂) ≦ 0, h₂(x₁, x₂) ≦ 0}. Then, . Let, i = 1, 2. Then, . Let α₁ = β₁ = α₂ = β₂ = 0, , , , , . Let u₁ = (-1, 0) u₂ = (0, 1), y₁ = (0, 0) and y₂ = (1, 0). Let, and. Letand. Then, , i = 1, 2, , i = 1, 2, , j = 1, 2, , k = 1, 2, and, k = 1, 2. Moreover,

and

Thus, Theorem 3.2 holds.

(2) Let. Then, is a weakly ε-efficient solution of (MFP)₂, but not an ε-efficient solution of (MFP)₂. Let. Then, clB = cone co{(0, -1, -1), (1, 0, 0), (0, 0, 1)}. However, (1, 0, 0) ∉ B. Thus, B is not closed. Moreover,, . Letand. Then,and. Let μ₁ = 1, μ₂ = 0, α₁ = β₁ = α₂ = β₂ = 0 and . Let, , , , n ∈ ℕ. Then,, , , , . Let u₁ = (-1, 0) and u₂ = y₁ = y₂ = (0, 0). Then,, , , . Letand. Then,and. Thus, , and. Hence, Theorem 3.4 holds.