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On εoptimality conditions for multiobjective fractional optimization problems
Fixed Point Theory and Applications volume 2011, Article number: 6 (2011)
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 nonfractional multiobjective convex optimization problem (NMCP) _{ v } with parametric v ∈ ℝ ^{p} , and then give the equivalent relation between (weakly) εefficient 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}  h_{0}(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 semicontinuous convex functions. Let ε ≧ 0. if, where ri domh_{ i } is the relative interior of domh_{ i }, then for all,
3 εoptimality 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 = (ε_{1}, ..., 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 εefficient 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 εefficient 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 εefficient solution of (MFP).
(ii)is anefficient 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
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 εefficient solution of (MFP).
(ii)is a weaklyefficient 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 εoptimality theorem for εefficient 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 εefficient 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  h_{0}(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 εoptimality theorem for εefficient solution of (MFP) which holds without any constraint qualification.
Theorem 3.2 Let. Suppose thatand, i = 1, ..., p. Thenis an εefficient 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 εefficient 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 εoptimality theorem for weakly εefficient 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 εefficient 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 εefficient 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 εoptimality theorem for weakly εefficient solution of (MFP) which holds without any constraint qualification.
Theorem 3.4 Letand assume that. Thenis a weakly εefficient 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 εefficient 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 f_{1}(x_{1}, x_{2}) = x_{1}, g_{1}(x_{1}, x_{2}) = 1, f_{2}(x_{1}, x_{2}) = x_{2}, g_{2}(x_{1}, x_{2}) = x_{1}, h_{1}(x_{1}, x_{2}) = x_{1}+ 1 and h_{2}(x_{1}, x_{2}) = x_{2} + 1.
(1)Let. Thenis an εefficient 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 α_{1} = β_{1} = γ_{1} = q_{1} = z_{1} = α_{2} = β_{2} = γ_{2} = q_{2} = z_{2} = 0, and let μ_{1} = μ_{2} = 1, and λ_{1} = 0 and λ_{1} = 2. Moreover, , , , , , , , , .
Thus,and.
Thus, (iii) of Theorem 3.1 holds.
(2) Let. Thenis not an εefficient solution of (MFP)_{1}, butis a weakly εefficient solution of (MFP)_{1}.
Let. Then
Hence, C is closed. Moreover,, and. Letand. Then,, . Let μ_{1} = 1 and μ_{2} = 1. Then,
Since (1, 0,1) ∈ C, . So, (ii) of Theorem 3.3 holds. Let α_{1} = β_{1} = γ_{1} = α_{2} = β_{2} = γ_{2} = 0, λ_{1} = 1 and λ_{2} = 0. Then,
and
Thus, (iii) of Theorem 3.3 holds.
Example 3.2 Consider the following MFP:
Let, and f_{1}(x_{1}, x_{2}) = x_{1} + 1, g_{1}(x_{1}, x_{2}) = 1, f_{2}(x_{1}, x_{2}) = x_{2}, g_{2}(x_{1}, x_{2}) = x_{1} + 1, h_{1}(x_{1}, x_{2}) = [max{0, x_{1}}]^{2}and h_{2}(x_{1}, x_{2}) = x_{2} + 1.
(1) Let. Then,is an εefficient 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 = {(x_{1}, x_{2}) ∈ ℝ ^{n}  h_{1}(x_{1}, x_{2}) ≦ 0, h_{2}(x_{1}, x_{2}) ≦ 0}. Then, . Let, i = 1, 2. Then, . Let α_{1} = β_{1} = α_{2} = β_{2} = 0, , , , , . Let u_{1} = (1, 0) u_{2} = (0, 1), y_{1} = (0, 0) and y_{2} = (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 εefficient solution of (MFP)_{2}, but not an εefficient 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 μ_{1} = 1, μ_{2} = 0, α_{1} = β_{1} = α_{2} = β_{2} = 0 and . Let, , , , n ∈ ℕ. Then,, , , , . Let u_{1} = (1, 0) and u_{2} = y_{1} = y_{2} = (0, 0). Then,, , , . Letand. Then,and. Thus, , and. Hence, Theorem 3.4 holds.
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Acknowledgements
This study was supported by the Korea Science and Engineering Foundation (KOSEF) NRL program grant funded by the Korea government(MEST)(No. ROA2008000200100).
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The authors declare that they have no competing interests.
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The authors, together discussed and solved the problems in the manuscript. All Authors read and approved the final manuscript.
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Kim, M.H., Kim, G.S. & Lee, G.M. On εoptimality conditions for multiobjective fractional optimization problems. Fixed Point Theory Appl 2011, 6 (2011). https://doi.org/10.1186/1687181220116
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DOI: https://doi.org/10.1186/1687181220116