# 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 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 [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 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 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 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.

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 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 .

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 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).

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 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) 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 . Then,, , , , . Let u1 = (-1, 0) and u2 = y1 = y2 = (0, 0). Then,, , , . Letand. Then,and. Thus, , and. Hence, Theorem 3.4 holds.

## References

1. 1.

Jeyakumar V, Lee GM, Dinh N: New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs. SIAM J Optim 2003,14(2):534–547.

2. 2.

Jeyakumar V, Wu ZY, Lee GM, Dinh N: Liberating the subgradient optimality conditions from constraint qualification. J Global Optim 2006,36(1):127–137.

3. 3.

Kim GS, Lee GM: On ε -approximate solutions for convex semidefinite optimization problems. Taiwanese J Math 2007,11(3):765–784.

4. 4.

Lee GM, Lee JH: ε -Duality theorems for convex semidefinite optimization problems with conic constraints. J Inequal 2010, 13. Art. ID363012

5. 5.

Kim GS, Lee GM: On ε -optimality theorems for convex vector optimization problems. To appear in Journal of Nonlinear and Convex Analysis

6. 6.

Govil MG, Mehra A: ε -Optimality for multiobjective programming on a Banach space. Eur J Oper Res 2004,157(1):106–112.

7. 7.

Gutiárrez C, Jimá B, Novo V: Multiplier rules and saddle-point theorems for Helbig's approximate solutions in convex Pareto problems. J Global Optim 2005,32(3):367–383.

8. 8.

Hamel A: An ε -Lagrange multiplier rule for a mathematical programming problem on Banach spaces. Optimization 2001,49(1–2):137–149.

9. 9.

Liu JC: ε -Duality theorem of nondifferentiable nonconvex multiobjective programming. J Optim Theory Appl 1991,69(1):153–167.

10. 10.

Liu JC: ε -Pareto optimality for nondifferentiable multiobjective programming via penalty function. J Math Anal Appl 1996,198(1):248–261.

11. 11.

Loridan P: Necessary conditions for ε -optimality. Optimality and stability in mathematical programming. Math Program Stud 1982, 19: 140–152.

12. 12.

Loridan P: ε -Solutions in vector minimization problems. J Optim Theory Appl 1984,43(2):265–276.

13. 13.

Strodiot JJ, Nguyen VH, Heukemes N: ε -Optimal solutions in nondifferentiable convex programming and some related questions. Math Program 1983,25(3):307–328.

14. 14.

Yokoyama K: Epsilon approximate solutions for multiobjective programming problems. J Math Anal Appl 1996,203(1):142–149.

15. 15.

Yokoyama K, Shiraishi S: ε -Necessary conditions for convex multiobjective programming problems without Slater's constraint qualification. , in press.

16. 16.

Hiriart-Urruty JB, Lemarechal C: Convex Analysis and Minimization Algorithms, vols. I and II. Springer-Verlag, Berlin; 1993.

17. 17.

Rockafellar RT: Convex Analysis. Princeton University Press, Princeton, NJ; 1970.

18. 18.

Zalinescu C: Convex Analysis in General Vector Space. World Scientific Publishing Co. Pte. Ltd, Singapore; 2002.

19. 19.

Jeyakumar V: Asymptotic dual conditions characterizing optimality for convex programs. J Optim Theory Appl 1997,93(1):153–165.

20. 20.

Jeyakumar V, Lee GM, Dinh N: Characterizations of solution sets of convex vector minimization problems. Eur. J Oper Res 2006,174(3):1380–1395.

21. 21.

Sawaragi Y, Nakayama H, Tanino T: Theory of Multiobjective Optimization. Academic Press, New York; 1985.

22. 22.

Gupta P, Shiraishi S, Yokoyama K: ε -Optimality without constraint qualification for multiobjective fractional problem. J Nonlinear Convex Anal 2005,6(2):347–357.

## 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).

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Authors

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Correspondence to Gue Myung Lee.

### 4 Competing interests

The authors declare that they have no competing interests.

### 5 Authors' contributions

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 ε-optimality conditions for multiobjective fractional optimization problems. Fixed Point Theory Appl 2011, 6 (2011). https://doi.org/10.1186/1687-1812-2011-6

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### Keywords

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