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On ε-optimality conditions for multiobjective fractional optimization problems

Fixed Point Theory and Applications20112011:6

https://doi.org/10.1186/1687-1812-2011-6

Received: 31 January 2011

Accepted: 21 June 2011

Published: 21 June 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 non-fractional multiobjective convex optimization problem (NMCP) v with parametric v 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.

Keywords

  • Weakly ε- efficient solution
  • ε- optimality condition
  • Multiobjective fractional optimization problem

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 [515] 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 [1618]. Let g : n {+∞} be a convex function. The subdifferential of g at a is given by
where domg: = {x n | g(x) < ∞} and ·, · is the scalar product on n . Let ε 0. The ε- subdifferential of g at a domg is defined by
The conjugate function of g : n {+∞} is defined by
The epigraph of g, epig, is defined by

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

Lemma 2.1[19]If h : n {+∞} is a proper lower semicontinuous convex function and if a domh, then
Lemma 2.2[20]Let h : n be a continuous convex function and u : n {+∞} 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 : n , i = 0, 1, , l be convex functions. Suppose that {x n | h i (x) 0, i = 1, , l} ≠ . Then the following statements are equivalent:

(i) {x n | h i (x) 0, i = 1, ..., l} {x n | h0(x) 0}

(ii) .

Proof. Let Q = {x 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 : n {+∞}, 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 : n , i = 1, ..., p be convex functions, g i : n , 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 : n , 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 point is 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 point is 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 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].
  1. (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 ,
  1. (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 Let and suppose that . Then the following are equivalent:

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

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

(iii) there exists such that
Proof. (i) (ii): The proof is also following the similar lines of Proposition 3.1.
  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.
  1. (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 Let and assume that and i = 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 .
  1. (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).
  1. (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 that and , i = 1, ..., p. Then is 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 Let and assume that , i = 1, ..., p, and is 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).
  1. (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 Let and assume that . Then is 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 . Then is an ε-efficient solution of (MFP)1.

Let and . Then , and
Thus, . It is clear that and . 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 . Then is not an ε-efficient solution of (MFP)1, but is a weakly ε-efficient solution of (MFP)1.

Let . Then
Hence, C is closed. Moreover, , and . Let and . 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) 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 . Let and . 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, , . Let and . Then, and . Let μ1 = 1, μ2 = 0, α1 = β1 = α2 = β2 = 0 and . Let , , , , n . Then, , , , , . Let u1 = (-1, 0) and u2 = y1 = y2 = (0, 0). Then, , , , . Let and . Then, and . Thus, , and . Hence, Theorem 3.4 holds.

Declarations

Acknowledgements

This study was supported by the Korea Science and Engineering Foundation (KOSEF) NRL program grant funded by the Korea government(MEST)(No. ROA-2008-000-20010-0).

Authors’ Affiliations

(1)
School of Free Major, Tongmyong University, Busan, Korea
(2)
Department of Applied Mathematics, Pukyong National University, Busan, Korea

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Copyright

© Kim et al; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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