# On *ε*-optimality conditions for multiobjective fractional optimization problems

- Moon Hee Kim
^{1}, - Gwi Soo Kim
^{2}and - Gue Myung Lee
^{2}Email author

**2011**:6

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

© Kim et al; licensee Springer. 2011

**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* ∈ 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.

## Keywords

*ε-*efficient solution

*ε-*optimality conditionMultiobjective 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 [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

*g*: ℝ

^{ n }→ ℝ ∪ {+∞} be a convex function. The subdifferential of

*g*at

*a*is given by

*g*: = {

*x*∈ ℝ

^{ n }|

*g*(

*x*) < ∞} and ⟨·, ·⟩ is the scalar product on ℝ

^{ n }. Let

*ε*≧ 0. The

*ε-*subdifferential of

*g*at

*a*∈ dom

*g*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*∈ dom

*h, 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}

*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 dom

*h*

_{ i }

*is the relative interior of*dom

*h*

_{ i }

*, then for all*,

## 3 *ε*-optimality theorems

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 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]).

_{ 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
.

Therefore, is an -efficient solution of , where .

*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*
.

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

- (ii)

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.

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

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

*(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*

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

- (ii)

*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*

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

*α*

_{ i }≧ 0, ,

*i*= 1, ...,

*p*,

*β*

_{ i }≧ 0, ,

*i*= 1, ...,

*p*, , , ,

*j*= 1, ...,

*m*, , , , , ,

*k*= 1, ...,

*p*, such that

*α*

_{ i }≧ 0, ,

*i*= 1, ...,

*p*,

*β*

_{ i }≧ 0, ,

*i*= 1, ...,

*p*, , , ,

*j*= 1, ...,

*m*, , , , , ,

*k*= 1, ...,

*p*, such that

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)*.

*(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*

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

- (ii)

⇔ (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*

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

*μ*

_{ i }≧ 0,

*i*= 1, ...,

*p*, ,

*α*

_{ i }≧ 0, ,

*i*= 1, ...,

*p*,

*β*

_{ i }≧ 0, ,

*i*= 1, ...,

*p*, , , ,

*j*= 1, ...,

*m*, such that

*μ*

_{ i }≧ 0,

*i*= 1, ...,

*p*, ,

*α*

_{ i }≧ 0, ,

*i*= 1, ...,

*p*,

*β*

_{ i }≧ 0, ,

*i*= 1, ...,

*p*, , , ,

*j*= 1, ...,

*m*, such that

□

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

*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*
. *Then*
*is an ε-efficient solution of* (MFP)_{1}.

*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, (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}.

*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*,

*Thus, (iii) of Theorem 3.3 holds*.

*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 ε-efficient solution of*(MFP)

_{2}.

*Let*.

*Then*, cl

*A*= cone co{(0, -1, -1), (1, 0, 0), (-1, 0, 0), (1, 1, 1), (0, 0, 1)}.

*Here*, (1, 0, 0) ∈ cl

*A*,

*but*(1, 0, 0) ∈

*A, where*cl

*A 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*.

*Let*

*and*.

*Then*, ,

*i*= 1, 2, ,

*i*= 1, 2, ,

*j*= 1, 2, ,

*k*= 1, 2,

*and*,

*k*= 1, 2.

*Moreover*,

*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*, cl*B* = 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 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,*
,
,
,
. *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

## References

- 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.MathSciNetView ArticleGoogle Scholar - Jeyakumar V, Wu ZY, Lee GM, Dinh N:
**Liberating the subgradient optimality conditions from constraint qualification.***J Global Optim*2006,**36**(1):127–137.MathSciNetView ArticleGoogle Scholar - Kim GS, Lee GM:
**On**ε**-approximate solutions for convex semidefinite optimization problems.***Taiwanese J Math*2007,**11**(3):765–784.MathSciNetGoogle Scholar - Lee GM, Lee JH: ε
**-Duality theorems for convex semidefinite optimization problems with conic constraints.***J Inequal*2010, 13. Art. ID363012Google Scholar - Kim GS, Lee GM: On ε -optimality theorems for convex vector optimization problems. To appear in Journal of Nonlinear and Convex AnalysisGoogle Scholar
- Govil MG, Mehra A: ε
**-Optimality for multiobjective programming on a Banach space.***Eur J Oper Res*2004,**157**(1):106–112.MathSciNetView ArticleGoogle Scholar - 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.MathSciNetView ArticleGoogle Scholar - Hamel A:
**An**ε**-Lagrange multiplier rule for a mathematical programming problem on Banach spaces.***Optimization*2001,**49**(1–2):137–149.MathSciNetView ArticleGoogle Scholar - Liu JC: ε
**-Duality theorem of nondifferentiable nonconvex multiobjective programming.***J Optim Theory Appl*1991,**69**(1):153–167.MathSciNetView ArticleGoogle Scholar - Liu JC: ε
**-Pareto optimality for nondifferentiable multiobjective programming via penalty function.***J Math Anal Appl*1996,**198**(1):248–261.MathSciNetView ArticleGoogle Scholar - Loridan P:
**Necessary conditions for**ε**-optimality. Optimality and stability in mathematical programming.***Math Program Stud*1982,**19:**140–152.MathSciNetView ArticleGoogle Scholar - Loridan P: ε
**-Solutions in vector minimization problems.***J Optim Theory Appl*1984,**43**(2):265–276.MathSciNetView ArticleGoogle Scholar - Strodiot JJ, Nguyen VH, Heukemes N: ε
**-Optimal solutions in nondifferentiable convex programming and some related questions.***Math Program*1983,**25**(3):307–328.MathSciNetView ArticleGoogle Scholar - Yokoyama K:
**Epsilon approximate solutions for multiobjective programming problems.***J Math Anal Appl*1996,**203**(1):142–149.MathSciNetView ArticleGoogle Scholar - Yokoyama K, Shiraishi S: ε -Necessary conditions for convex multiobjective programming problems without Slater's constraint qualification. , in press.Google Scholar
- Hiriart-Urruty JB, Lemarechal C:
*Convex Analysis and Minimization Algorithms, vols. I and II.*Springer-Verlag, Berlin; 1993.Google Scholar - Rockafellar RT:
*Convex Analysis.*Princeton University Press, Princeton, NJ; 1970.View ArticleGoogle Scholar - Zalinescu C:
*Convex Analysis in General Vector Space.*World Scientific Publishing Co. Pte. Ltd, Singapore; 2002.Google Scholar - Jeyakumar V:
**Asymptotic dual conditions characterizing optimality for convex programs.***J Optim Theory Appl*1997,**93**(1):153–165.MathSciNetView ArticleGoogle Scholar - Jeyakumar V, Lee GM, Dinh N:
**Characterizations of solution sets of convex vector minimization problems. Eur.***J Oper Res*2006,**174**(3):1380–1395.MathSciNetView ArticleGoogle Scholar - Sawaragi Y, Nakayama H, Tanino T:
*Theory of Multiobjective Optimization.*Academic Press, New York; 1985.Google Scholar - Gupta P, Shiraishi S, Yokoyama K: ε
**-Optimality without constraint qualification for multiobjective fractional problem.***J Nonlinear Convex Anal*2005,**6**(2):347–357.MathSciNetGoogle Scholar

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