RESEARC H Open Access On ε-optimality conditions for multiobjective fractional optimization problems Moon Hee Kim 1 , Gwi Soo Kim 2 and Gue Myung Lee 2* * Correspondence: gmlee@pknu.ac. kr 2 Department of Applied Mathematics, Pukyong National University, Busan 608-737, Korea Full list of author information is available at the end of the article 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 Î ℝ p , and then give the equivalent relation between (weakly) ε- efficient solution of (MFP) and (weakly) ¯ ε -efficient solution of ( NMCP ) ¯ v . 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 opti- mization problems to obtain optimality co nditions 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 impo rtant to investigate an optimality condition for an efficient solution o f 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 theo- rems 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 multi- objective optimization problems, which consist of more than two objective functions and a constrained set. Kim et al. Fixed Point Theory and Applications 2011, 2011:6 http://www.fixedpointtheoryandapplications.com/content/2011/1/6 © 2011 Kim et al; licensee Springer. 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 reprodu ction in any medium, provided the original work is properly cited. 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 qual ifica- tions. 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 subdif ferent ial of g at a is given by ∂g ( a ) := {v ∈ R n | g ( x ) g ( a ) + v, x − a, ∀x ∈ domg} , where domg:={x Î ℝ n | g(x)<∞}and〈·, ·〉 is the scalar product on ℝ n .Letε ≧ 0. The ε-subdifferential of g at a Î domg is defined by ∂ ε g ( a ) := {v ∈ R n | g ( x ) g ( a ) + v, x − a−ε, ∀x ∈ domg} . The conjugate function of g : ℝ n ® ℝ ∪ {+∞} is defined by g ∗ ( v ) =sup{v, x−g ( x ) | x ∈ R n } . The epigraph of g, epig, is defined by epig = { ( x, r ) ∈ R n × R | g ( x ) r} . For a nonempty closed convex set C ⊂ ℝ n , δ C : ℝ n ® ℝ ∪ {+∞} is called the indicator of C if δ C (x)= 0ifx ∈ C, +∞ otherwis e . Lemma 2.1 [19]If h : ℝ n ® ℝ ∪ {+∞} is a proper lower semicontinuous convex func- tion and if a Î domh, then epih ∗ = ε 0 {(v, v, a + ε − h(a))|v ∈ ∂ ε h(a)} . 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 epi ( h + u ) ∗ =epih ∗ +epiu ∗ . 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 Î ℝ n | h i (x) ≦ 0, i = 1, , l} ≠ ∅. Then the following statements are equivalent: (i) {x Î ℝ n | h i (x) ≦ 0, i = 1, , l} ⊆ {x Î ℝ n | h 0 (x) ≧ 0} (ii) 0 ∈ epih ∗ 0 +cl λ i 0 epi( l i=1 λ i h i ) ∗ . Kim et al. Fixed Point Theory and Applications 2011, 2011:6 http://www.fixedpointtheoryandapplications.com/content/2011/1/6 Page 2 of 13 Proof. Let Q ={x Î ℝ n | h i (x) ≦ 0, i = 1, , l}. Then Q ≠ ∅ and by Lemma 2.1 in [2], epiδ ∗ Q =cl λ i 0 epi( l i=1 λ i h i ) ∗ . 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 funct ions. Let ε ≧ 0. if m i =1 ri domh i = 0 , where ri domh i is the relative interior of domh i , then for all x ∈ m i =1 domh i , ∂ ε ( m i =1 h i )(x)= { m i =1 ∂ ε i h i (x) | ε i 0, i =1,··· , m, m i =1 ε i = ε} . 3 ε-optimality theorems Consider the following MFP: (MFP) Minimize f (x) g(x) := f 1 (x) g 1 (x) , ··· , f p (x) g p (x) subject to x ∈ Q := {x ∈ R n |h j (x) 0, j =1, , m} . 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 Î 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 ¯ x ∈ Q is said to be an ε-efficient solution of (MFP) if there does not exist x Î Q such that f i (x) g i (x) f i ( ¯ x) g i ( ¯ x) − ε i ,foralli =1, , p, f j (x) g j (x) < f j ( ¯ x) g j ( ¯ x) − ε j ,forsomej ∈{1, , p} . When ε = 0, t hen the ε-efficiency becomes the efficiency for (MFP) (see the defini- tion 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 Apoint ¯ x ∈ Q is said to be a weakly ε-efficient solution of (MFP) if there does not exist x Î Q such that f i (x) g i ( x ) < f i ( ¯ x) g i ( ¯ x ) − ε i ,foralli =1, , p . 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 transf orm the problem (MFP) into the nonfr actional multiobjective convex optimization problem (NMCP) v with parametric v Î ℝ p : (NMCP) v Minimize (f (x) − vg(x)) := (f 1 (x) − v 1 g 1 (x), , f p (x) − v p g p (x) ) sub j ect to x ∈ Q. Kim et al. Fixed Point Theory and Applications 2011, 2011:6 http://www.fixedpointtheoryandapplications.com/content/2011/1/6 Page 3 of 13 Adapting Lemma 4.1 in [22] and modifying Proposition 3.1 in [12], we can obtain the following proposition: Proposition 3.1 Let ¯ x ∈ Q . Then the following are equivalent: (i) ¯ x is an ε-efficient solution of (MFP). (ii) ¯ x is an ¯ ε -efficient solution of ( NMCP ) ¯ v ,where ¯ v := f 1 ( ¯ x) g 1 ( ¯ x) − ε 1 , , f p ( ¯ x) g p ( ¯ x) − ε p and ¯ε =(ε 1 g 1 ( ¯ x), , ε p g p ( ¯ x) ) . (iii) Q ∩ S ( ¯ x ) = ∅ or p i=1 f i (x) − f i ( ¯ x) g i ( ¯ x) − ε i g i (x) 0= p i=1 f i ( ¯ x) − f i ( ¯ x) g i ( ¯ x ) − ε i g i ( ¯ x) − p i=1 ε i g i ( ¯ x) for any x ∈ Q ∩ S( ¯ x) , where S( ¯ x)={x ∈ R n | f i (x)− f i ( ¯ x) g i ( ¯ x) − ε i g i (x) 0=f i ( ¯ x)− f i ( ¯ x) g i ( ¯ x) − ε i g i ( ¯ x)−¯ε i , i =1, , p } . Proof. (i) ⇔ (ii): It follows from Lemma 4.1 in [22]. (ii) ⇒ (iii): Let ¯ x be an ¯ ε -efficient solution of ( NMCP ) ¯ v ,where ¯ v := f 1 ( ¯ x) g 1 ( ¯ x) − ε 1 , , f p ( ¯ x) g p ( ¯ x) − ε p and ¯ε =(ε 1 g 1 ( ¯ x), , ε p g p ( ¯ x) ) .Then Q ∩ S ( ¯ x ) = ∅ or Q ∩ S ( ¯ x ) = ∅ . Suppose that Q ∩ S ( ¯ x ) = ∅ . Then for any x ∈ Q ∩ S ( ¯ x ) and all i =1, p, f i (x) − f i ( ¯ x) g i ( ¯ x ) − ε i g i (x) f i ( ¯ x) − f i ( ¯ x) g i ( ¯ x ) − ε i g i ( ¯ x) −¯ε i . Hence the ¯ ε -efficiency of ¯ x yields f i (x) − f i ( ¯ x) g i ( ¯ x ) − ε i g i (x)=f i ( ¯ x) − f i ( ¯ x) g i ( ¯ x ) − ε i g i ( ¯ x) −¯ε i for any x ∈ Q ∩ S ( ¯ x ) and all i = 1, , p. Thus we have, for all x ∈ Q ∩ S ( ¯ x ) , p i =1 f i (x) − f i ( ¯ x) g i ( ¯ x) − ε i g i (x) = p i =1 f i ( ¯ x) − f i ( ¯ x) g i ( ¯ x) − ε i g i ( ¯ x) − p i =1 ¯ε i . (iii) ⇒ (ii): Suppose that Q ∩ S ( ¯ x ) = ∅ . Then there does not exist x Î Q such that x ∈ S ( ¯ x ) ; that is, there does not exist x Î Q such that f i (x) − f i ( ¯ x) g i ( ¯ x ) − ε i g i (x) f i ( ¯ x) − f i ( ¯ x) g i ( ¯ x ) − ε i g i ( ¯ x) −¯ε i for all i = 1, , p. Hence, there does not exist x Î Q such that f i (x) − f i ( ¯ x) g i ( ¯ x) − ε i g i (x) f i ( ¯ x) − f i ( ¯ x) g i ( ¯ x) − ε i g i ( ¯ x) −¯ε i , i =1, , p, f j (x) − f j ( ¯ x) g j ( ¯ x) − ε j g j (x) < f j ( ¯ x) − f j ( ¯ x) g j ( ¯ x) − ε j g j ( ¯ x) −¯ε j ,forsomej ∈{1, , p} . Therefore, ¯ x is an ¯ ε -efficient solution of ( NMCP ) ¯ v ,where ¯ v := f 1 ( ¯ x) g 1 ( ¯ x) − ε 1 , , f p ( ¯ x) g p ( ¯ x) − ε p . Kim et al. Fixed Point Theory and Applications 2011, 2011:6 http://www.fixedpointtheoryandapplications.com/content/2011/1/6 Page 4 of 13 Assume that Q ∩ S ( ¯ x ) = ∅ . Then, from this assumption p i =1 f i (x) − f i ( ¯ x) g i ( ¯ x) − ε i g i (x) p i =1 f i ( ¯ x) − f i ( ¯ x) g i ( ¯ x) − ε i g i ( ¯ x) − p i =1 ¯ε i , (3:1) for any x ∈ Q ∩ S ( ¯ x ) . Suppose to the contrary that ¯ x is not an ¯ ε -efficient solution of ( NMCP ) ¯ v . Then, there exist ˆ x ∈ Q and an index j such that f i ( ˆ x) − ¯ v i g i ( ˆ x) f i ( ¯ x) − ¯ vg i ( ¯ x) −¯ε i , i =1, , p, f j ( ˆ x) − ¯ v j g j ( ˆ x) < f j ( ¯ x) − ¯ v j g j ( ¯ x) −¯ε j ,forsomej ∈{1, , p} . Therefore, ˆ x ∈ Q ∩ S ( ¯ x ) and p i=1 f i ( ˆ x) − f i ( ¯ x) g i ( ¯ x) − ε i g i ( ˆ x) < p i=1 f i ( ¯ x) − f i ( ¯ x) g i ( ¯ x) − ε i g i ( ¯ x) − p i=1 ¯ε i , which contradicts the above inequality. Hence, ¯ x is an ¯ ε -efficient solution of ( NMCP ) ¯ v . We can easily obtain the following proposition: Proposition 3.2 Let ¯ x ∈ Q and suppose that f i ( ¯ x ) ε i g i ( ¯ x ) , i =1, , p . Then the fol- lowing are equivalent: (i) ¯ x is a weakly ε-efficient solution of (MFP). (ii) ¯ x is a weakly ¯ ε -efficient solution of ( NMCP ) ¯ v , where ¯ε =(ε 1 g 1 ( ¯ x), , ε p g p ( ¯ x) ) and ¯ε =(ε 1 g 1 ( ¯ x), , ε p g p ( ¯ x) ) . (iii) there exists ¯ λ := ( ¯ λ 1 , , ¯ λ p ) ∈ R p + \{0 } such that p i=1 ¯ λ i f i (x) − f i ( ¯ x) g i ( ¯ x) − ε i g i (x) 0= p i =1 ¯ λ i f i ( ¯ x) − f i ( ¯ x) g i ( ¯ x) − ε i g i ( ¯ x) − p i =1 ¯ λ i ε i g i ( ¯ x) for any x ∈ Q . Proof. (i) ⇔ (ii): The proof is also following the similar lines of Proposition 3.1. (ii) ⇒ (iii): Let (x)=( 1 (x), , p (x)), ∀x Î Q,where ϕ i (x)=f i ( ¯ x) − f i ( ¯ x) g i ( ¯ x) − ε i g i (x), i =1,··· , p .Then, i (x), i = 1, , p, are convex. Since ¯ x ∈ Q is a weakly ε-efficient solution of ( NMCP ) ¯ v ,where ( ϕ ( Q ) + R p + ) ∩ ( −intR p + ) = ∅ , ( ϕ ( Q ) + R p + ) ∩ ( −intR p + ) = ∅ , and hence, it follows from separation theorem that there exist ¯ λ i 0 , i = 1, , p, ( ¯ λ 1 , , ¯ λ p ) = 0 such that p i =1 ¯ λ i ϕ i (x) 0 ∀x ∈ Q . Thus (iii) holds. (iii) ⇒ (ii): If (ii) does not hold, that is, ¯ x is not a weakly ¯ ε -efficient solution of ( NMCP ) ¯ v , 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 ¯ x ∈ Q and assume that Q ∩ S ( ¯ x ) = ∅ and f i ( ¯ x ) ε i g i ( ¯ x ) , i =1, , p i = 1, , p. Suppose that λ j 0 m j=1 epi(λ j h j ) ∗ + μ i 0 p i=1 epi(μ i f i ) ∗ +epi(− ¯ v i μ i g i ) ∗ is closed, where ¯ v i = f i ( ¯ x) g i ( ¯ x) − ε i , i = 1, , p. Then the following are equivalent. Kim et al. Fixed Point Theory and Applications 2011, 2011:6 http://www.fixedpointtheoryandapplications.com/content/2011/1/6 Page 5 of 13 (i) ¯ x is an ε-efficient solution of (MFP). (ii) 0 0 T ∈ p i=1 epif ∗ i +epi(− ¯ v i g i ) ∗ + λ j 0 m j=1 epi(λ j h j ) ∗ + μ i 0 p i=1 epi(μ i f i ) ∗ +epi(− ¯ v i μ i g i ) ∗ . (iii) there exist a i ≧ 0, u i ∈ ∂ α i f i ( ¯ x ) , b i ≧ 0, y i ∈ ∂ β i (− ¯ v i μ i g i )( ¯ x ) , i = 1, , p, l j ≧ 0, g j ≧ 0, w j ∈ ∂ γ j (λ j h j )( ¯ x) , j = 1, , m, μ i ≧ 0, q i ≧ 0, s i ∈ ∂ q i (μ i f i )( ¯ x ) , z i ≧ 0, t i ∈ ∂ z i (− ¯ v i μ i g i )( ¯ x ) i = 1, , p such that 0= p i=1 (u i + y i )+ m j =1 w j + p i=1 (s i + t i ) and p i=1 (α i + β i + q i + z i )+ m j =1 γ j = p i=1 ε i (1 + μ i )g i ( ¯ x)+ m j =1 λ j h j ( ¯ x) . Proof. Let h 0 (x)= p i =1 f i (x) − ¯ v i g i (x) . (i) ⇔ (by Proposition 3.1) h 0 (x) ≧ 0, ∀x ∈ Q ∩ S ( ¯ x ) . ⇔ {x|f i ( x ) − ¯ v i g i ( x ) 0 , i = 1, , p, h j (x) ≦ 0, j = 1, , m} ⊂ {x | h 0 (x) ≧ 0}. ⇔ (by lemma 2.3) 0 0 T ∈ p i=1 epif ∗ i +epi(− ¯ v i g i ) ∗ +cl ⎧ ⎨ ⎩ λ j 0 m j=1 epi(λ j h j ) ∗ + μ i 0 p i=1 epi(μ i f i ) ∗ +epi(− ¯ v i μ i g i ) ∗ ⎫ ⎬ ⎭ . Thus by the closedness assumption, (i) is equivalent to (ii). (ii) ⇔ (iii): (ii) ⇔ (by Lemma 2.1), there exist a i ≧ 0, u i ∈ ∂ α i (μ i f i )( ¯ x ) , i = 1, , p, b i ≧ 0, y i ∈ ∂ β i (− ¯ v i μ i g i )( ¯ x ) , i = 1, , p, l j ≧ 0, g j ≧ 0, w j ∈ ∂ γ j (λ j h j )( ¯ x) , j =1, ,m, μ i ≧ 0, q i ≧ 0, s i ∈ ∂ q i (μ i f i )( ¯ x ) , i = 1, , p, z i ≧ 0, t i ∈ ∂ z i (− ¯ v i μ i g i )( ¯ x ) , i = 1, , p such that 0 0 T = p i=1 u i u i , ¯ x + α i − f i ( ¯ x) T + y i y i , ¯ x + β i − (− ¯ v i g i )( ¯ x) T + m j=1 w j w j , ¯ x + γ j − (λ j h j )( ¯ x) T + p i=1 s i s i , ¯ x + q i − (μ i f i )( ¯ x) T + t i t i , ¯ x + z i − (− ¯ v i μ i g i )( ¯ x) T . ⇔ there exist a i ≧ 0, u i ∈ ∂ α i (μ i f i )( ¯ x ) , b i ≧ 0, y i ∈ ∂ β i (− ¯ v i μ i g i )( ¯ x ) , i = 1, , p, l j ≧ 0, g j ≧ 0, w j ∈ ∂ γ j (λ j h j )( ¯ x ) , j = 1, , m, μ i ≧ 0, q i ≧ 0, s i ∈ ∂ q i (μ i f i )( ¯ x ) , z i ≧ 0, t i ∈ ∂ z i (− ¯ v i μ i g i )( ¯ x ) i = 1, , p such that Kim et al. Fixed Point Theory and Applications 2011, 2011:6 http://www.fixedpointtheoryandapplications.com/content/2011/1/6 Page 6 of 13 0= p i=1 (u i + y i )+ m j =1 w j + p i=1 (s i + t i ) and p i=1 (α i + β i + q i + z i )+ m j=1 γ j = p i=1 ⎡ ⎣ f i ( ¯ x) − ¯ v i g i ( ¯ x)+(μ i f i )( ¯ x) − ( ¯ v i μ i g i )( ¯ x)+ m j=1 λ j h j ( ¯ x) ⎤ ⎦ . ⇔ (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 ¯ x ∈ Q . Suppose that Q ∩ S ( ¯ x ) = ∅ and f i ( ¯ x ) ε i g i ( ¯ x ) , i =1, , p , i = 1, , p. Then ¯ x is an ε-efficient solution of (MFP) if and only if there exist a i ≧ 0, u i ∈ ∂ α i (μ i f i )( ¯ x ) , i =1, ,p, b i ≧ 0, y i ∈ ∂ β i (− ¯ v i μ i g i )( ¯ x ) , i =1, ,p, λ n j 0 , γ n j 0 , w n j ∈ ∂ γ n j (λ n j h j )( ¯ x ) , j = 1, , m, μ n k 0 , q n k 0 , s n k ∈ ∂ q n k (μ n k f k )( ¯ x ) , z n k 0 , t n k ∈ ∂ z n k (− ¯ v k μ n k g k )( ¯ x ) , k = 1, , p such that 0= p i=1 (u i + y i ) + lim n→∞ ⎡ ⎣ m j=1 w n j + p k=1 (s n k + t n k ) ⎤ ⎦ and p i=1 ε i g i ( ¯ x)= p i=1 (α i + β i ) + lim n→∞ ⎧ ⎨ ⎩ m j=1 γ n j − (λ n j h j )( ¯ x) + p k =1 q n k + z n k − μ n k ε k g k ( ¯ x) . Proof. ¯ x is an ε-efficient solution of (MFP) ⇔ (from the proof of Theorem 3.1) 0 0 T ∈ p i=1 epif ∗ i +epi(− ¯ v i g i ) ∗ +cl ⎧ ⎨ ⎩ λ j 0 m j=1 epi(λ j h j ) ∗ + μ i 0 p i=1 epi(μ i f i ) ∗ +epi(− ¯ v i μ i g i ) ∗ ⎫ ⎬ ⎭ . ⇔ (by Lemma 2.1) there exist a i ≧ 0, u i ∈ ∂ α i (μ i f i )( ¯ x ) , i =1, ,p , b i ≧ 0, y i ∈ ∂ β i (− ¯ v i μ i g i )( ¯ x ) , i =1, ,p, λ n j 0 , γ n j 0 , w n j ∈ ∂ γ n j (λ n j h j )( ¯ x ) , j =1, ,m, μ n k 0 , s n k ∈ ∂ q n k (μ n k f k )( ¯ x ) , s n k ∈ ∂ q n k (μ n k f k )( ¯ x ) , z n k 0 , t n k ∈ ∂ z n k (− ¯ v k μ n k g k )( ¯ x ) , k = 1, , p, such that 0 0 T = p i=1 u i u i , ¯ x + α i − f i ( ¯ x) T + y i y i , ¯ x + β i − (− ¯ v i g i )( ¯ x) T + lim n→∞ ⎧ ⎨ ⎩ m j=1 w n j w n j , ¯ x + γ n j − (λ n j h j )( ¯ x) T + p k =1 s n k s n k , ¯ x + q n k − (μ n k f k )( ¯ x) T + t n k t n k , ¯ x + z n k − (− ¯ v k μ n k g i )( ¯ x) T . Kim et al. Fixed Point Theory and Applications 2011, 2011:6 http://www.fixedpointtheoryandapplications.com/content/2011/1/6 Page 7 of 13 ⇔ there exist a i ≧ 0, u i ∈ ∂ α i (μ i f i )( ¯ x ) , i = 1, , p, b i ≧ 0, y i ∈ ∂ β i (− ¯ v i μ i g i )( ¯ x ) , i = 1, , p, λ n j 0 , γ n j 0 , w n j ∈ ∂ γ n j (λ n j h j )( ¯ x ) , j =1, ,m, μ n k 0 , q n k 0 , s n k ∈ ∂ q n k (μ n k f k )( ¯ x ) , t n k ∈ ∂ z n k (− ¯ v k μ n k g k )( ¯ x ) , t n k ∈ ∂ z n k (− ¯ v k μ n k g k )( ¯ x ) , k = 1, , p, such that 0= p i=1 (u i + y i ) + lim n→∞ ⎡ ⎣ m j=1 w n j + p k=1 (s n k + t n k ) ⎤ ⎦ and p i=1 ε i g i ( ¯ x)= p i=1 (α i + β i ) + lim n→∞ ⎧ ⎨ ⎩ m j=1 γ n j − (λ n j h j )( ¯ x) + p k =1 q n k + z n k − μ n k ε k g k ( ¯ x) . We present a necessary and sufficient ε-optimality theorem for weakly ε-efficient solution of (MFP) under a constraint qualification. Theorem 3.3 Let ¯ x ∈ Q and assume that f i ( ¯ x ) ε i g i ( ¯ x ) , i =1, , p , i = 1, , p, and λ j 0 m j=1 epi(λ j h j ) ∗ is closed. Then the following are equivalent. (i) ¯ x is a weakly ε-efficient solution of (MFP). (ii) there exist μ i ≧ 0, i = 1, , p, p i =1 μ i = 1 such that 0 0 T ∈ p i=1 epi(μ i f i ) ∗ +epi(− ¯ v i μ i g i ) ∗ + λ j 0 m j=1 epi(λ j h j ) ∗ , where ¯ v i = f i ( ¯ x) g i ( ¯ x ) − ε i , i = 1, , p. (iii) there exist μ i ≧ 0, p i =1 μ i = 1 , a i ≧ 0, u i ∈ ∂ α i (μ i f i )( ¯ x ) , b i ≧ 0, y i ∈ ∂ β i (− ¯ v i μ i g i )( ¯ x ) , i = 1, , p, l j ≧ 0, g j ≧ 0, w j ∈ ∂ γ j (λ j h j )( ¯ x ) , j = 1, , m, such that 0= p i=1 (u i + y i )+ m j =1 w j and p i=1 μ i ε i g i ( ¯ x)= p i=1 (α i + β i )+ m j =1 γ j − (λ j h j )( ¯ x) . Proof. (i) ⇔ (ii): ¯ x is a weakly ε-efficient solution of (MFP) ⇔ (by Proposition 3.2) there exist μ i ≧ 0, i = 1, , p, p i =1 μ i = 1 such that p i =1 μ i [f i (x) − ¯ v i g i (x)] 0 ∀x ∈ Q ⇔ there exist μ i ≧ 0, i = 1, , p, p i =1 μ i = 1 such that {x|h j (x) 0, j =1, , m}⊂{x| p i =1 μ i f i (x) − ¯ v i g i (x) 0 } Kim et al. Fixed Point Theory and Applications 2011, 2011:6 http://www.fixedpointtheoryandapplications.com/content/2011/1/6 Page 8 of 13 ⇔ (by Lemma 2.3) there exist μ i ≧ 0, i = 1, , p, p i =1 μ i = 1 such that 0 0 T ∈ p i=1 epi(μ i f i ) ∗ +epi(− ¯ v i μ i g i ) ∗ +cl ⎧ ⎨ ⎩ λ j 0 m j=1 epi(λ j h j ) ∗ ⎫ ⎬ ⎭ . Thus, by the closedness assumption, (i) is equivalent to (ii). (ii) ⇔ (iii): (ii) ⇔ (by Lemma 2.1) there exist μ i ≧ 0, p i =1 μ i = 1 , a i ≧ 0, u i ∈ ∂ α i (μ i f i )( ¯ x ) , b i ≧ 0, y i ∈ ∂ β i (− ¯ v i μ i g i )( ¯ x ) , i = 1, , p, l j ≧ 0, g j ≧ 0, w j ∈ ∂ γ j (λ j h j )( ¯ x ) , j = 1, , m, such that 0 0 T = p i=1 u i u i , ¯ x + α i − (μ i f i )( ¯ x) T + y i y i , ¯ x + β i − (− ¯ v i μ i g i )( ¯ x) T + m j =1 w j w j , ¯ x + γ j − (λ j h j )( ¯ x) T . ⇔ (iii) holds. □ Now, we propose a n ecessary and sufficient ε-optimality theorem for weakly ε-effi- cient solution of (MFP) which holds without any constraint qualification. Theorem 3.4 Let ¯ x ∈ Q and assume that f i ( ¯ x ) ε i g i ( ¯ x ) , i =1, , p . Then ¯ x is a weakly ε-efficient solution of (MFP) if and only if there exist μ i ≧ 0, i =1, ,p, p i =1 μ i = 1 , a i ≧ 0, u i ∈ ∂ α i (μ i f i )( ¯ x ) , i =1, ,p, b i ≧ 0, y i ∈ ∂ β i (− ¯ v i μ i g i )( ¯ x ) , i = 1, , p, γ n j 0 , γ n j 0 , w n j ∈ ∂ γ n j (λ n j h j )( ¯ x ) , j = 1, , m, such that 0= p i=1 (u i + y i ) + lim n→∞ m j =1 w n j and p i=1 μ i ε i g i ( ¯ x)= p i=1 (α i + β i ) + lim n→∞ m j =1 γ n j − (λ n j h j )( ¯ x) . Proof. ¯ x is a weakly ε-efficient solution of (MFP) ⇔ ((from the proof of Theorem 3.3) there e xist μ i ≧ 0, i =1, ,p, p i =1 μ i = 1 such that 0 0 T ∈ p i=1 epi(μ i f i ) ∗ +epi(− ¯ v i μ i g i ) ∗ +cl ⎧ ⎨ ⎩ λ j 0 m j=1 epi(λ j h j ) ∗ ⎫ ⎬ ⎭ . ⇔ (by Lemma 2.1) there exist μ i ≧ 0, i = 1, , p, p i =1 μ i = 1 , a i ≧ 0, u i ∈ ∂ α i (μ i f i )( ¯ x ) , i = 1, , p, b i ≧ 0, y i ∈ ∂ β i (− ¯ v i μ i g i )( ¯ x ) , i =1, ,p, λ n j 0 , γ n j 0 , w n j ∈ ∂ γ n j (λ n j h j )( ¯ x ) , j = 1, , m, such that 0 0 T = p i=1 u i u i , ¯ x + α i − (μ i f i )( ¯ x) T + y i y i , ¯ x + β i − (− ¯ v i μ i g i )( ¯ x) T + lim n→∞ ⎧ ⎨ ⎩ m j=1 w n j w n j , ¯ x + γ n j − (λ n j h j )( ¯ x) T ⎫ ⎬ ⎭ . Kim et al. Fixed Point Theory and Applications 2011, 2011:6 http://www.fixedpointtheoryandapplications.com/content/2011/1/6 Page 9 of 13 ⇔ there exist μ i ≧ 0, i =1, ,p, p i =1 μ i = 1 , a i ≧ 0, u i ∈ ∂ α i (μ i f i )( ¯ x ) , i = 1, , p, b i ≧ 0, y i ∈ ∂ β i (− ¯ v i μ i g i )( ¯ x ) , i = 1, , p, λ n j 0 , γ n j 0 , w n j ∈ ∂ γ n j (λ n j h n j )( ¯ x ) , j = 1, , m,such that 0= p i=1 (u i + y i ) + lim n→∞ m j =1 w n j and p i=1 μ i ε i g i ( ¯ x)= p i=1 (α i + β i ) + lim n→∞ m j =1 γ n j − (γ n j h j )( ¯ x) . □ Now, we give examples illustrating Theorems 3.1, 3.2, 3.3, and 3.4. Example 3.1 Consider the following MFP: (MFP) 1 Minimize x 1 , x 2 x 1 subject to ( x 1 , x 2 ) ∈ Q := { ( x 1 , x 2 ) ∈ R 2 |−x 1 +1 0, −x 2 +1 0} . Let ε =(ε 1 , ε 2 )=( 1 2 , 1 2 ) , 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 ( ¯ x 1 , ¯ x 2 )=( 3 2 , 9 4 ) . Then ( ¯ x 1 , ¯ x 2 ) is an ε-efficient solution of (MFP) 1 . Let ¯ v 1 = f 1 ( ¯ x 1 , ¯ x 2 ) g 1 ( ¯ x 1 , ¯ x 2 ) − ε 1 and ¯ v 2 = f 2 ( ¯ x 1 , ¯ x 2 ) g 2 ( ¯ x 1 , ¯ x 2 ) − ε 2 . Then ¯ v 1 = ¯ v 2 = 1 , and Q ∩ S ( ¯ x 1 , ¯ x 2 ) = Q ∩{( ¯ x 1 , ¯ x 2 ) ∈ R 2 |f 1 ( ¯ x 1 , ¯ x 2 ) − ¯ v 1 g 1 ( ¯ x 1 , ¯ x 2 ) 0, f 2 ( ¯ x 1 , ¯ x 2 ) − ¯ v 2 g 2 ( ¯ x 1 , ¯ x 2 ) 0 } = { ( 1, 1 ) }. Thus, Q ∩ S ( ¯ x 1 , ¯ x 2 ) = ∅ . It is clear that f 1 ( ¯ x 1 , ¯ x 2 ) ε 1 g 1 ( ¯ x 1 , ¯ x 2 ) and f 2 ( ¯ x 1 , ¯ x 2 ) ε 2 g 2 ( ¯ x 1 , ¯ x 2 ) . Let A = λ 1 ≥0, λ 2 ≥0 2 j=1 epi(λ j h j ) ∗ + μ 1 ≥0, μ 2 ≥0 2 j=1 [epi(μ j f j ) ∗ +epi(− ¯ v i μ i g i ) ∗ ] . Then A = λ 1 ≥0, λ 2 ≥0 μ 1 ≥0, μ 2 ≥0 epi ⎛ ⎝ 2 j=1 λ j h j + 2 i=1 μ i (f i − ¯ v i g i ) ⎞ ⎠ ∗ =coneco{ ( −1, 0, −1 ) , ( 0, −1, −1 ) , ( 1, 0, 1 ) , ( −1, 1, 0 ) , ( 0, 0, 1 ) } , wherecoDistheconvexhullofasetDand cone coD is the cone generated by coD. Thus A is closed. Let B = 2 i=1 [epif ∗ i +epi(− ¯ v i g i ) ∗ ]+ A . Then B = {(1, 0)} × [0, ∞)+{(0, 0)} × [1, ∞)+{(0, 1)} × [0, ∞)+{(-1, 0)} × [0, ∞)+A. Since (0,- 1,-1) Î A, (0, 0, 0) Î B. Thus (ii) of Theorem 3.1 holds. Let a 1 = b 1 = g 1 = q 1 = z 1 = a 2 = b 2 = g 2 = q 2 = z 2 =0,and let μ 1 = μ 2 =1,and l 1 =0and l 1 =2.Moreover, ∂f 2 ( ¯ x 1 , ¯ x 2 ) = { ( 0, 1 )} , ∂f 2 ( ¯ x 1 , ¯ x 2 ) = { ( 0, 1 )} , ∂ ( − ¯ v 1 g 1 )( ¯ x 1 , ¯ x 2 ) = { ( 0, 0 )} , ∂ ( − ¯ v 2 g 2 )( ¯ x 1 , ¯ x 2 ) = { ( −1, 0 )} , ∂ ( λ 2 h 2 )( ¯ x 1 , ¯ x 2 ) = { ( 0, −2 ) } , , ∂ ( λ 2 h 2 )( ¯ x 1 , ¯ x 2 ) = { ( 0, −2 ) } , ∂ ( μ 1 f 1 )( ¯ x 1 , ¯ x 2 ) = { ( 1, 0 )} , ∂ ( − ¯ v 1 μ 1 g 1 )( ¯ x 1 , ¯ x 2 ) = { ( 0, 0 )} , ∂ ( − ¯ v 1 μ 1 g 1 )( ¯ x 1 , ¯ x 2 ) = { ( 0, 0 )} , ∂ ( − ¯ v 2 μ 2 g 2 )( ¯ x 1 , ¯ x 2 ) = { ( −1, 0 )} . Thus, 2 i =1 ∂(f i − ¯ v i g i )( ¯ x 1 , ¯ x 2 )+ 2 i =1 ∂(λ i h i )( ¯ x 1 , ¯ x 2 )+ 2 i =1 ∂(μ i f i − ¯ v i μ i g i )( ¯ x 1 , ¯ x 2 )={(0, 0) } and 2 i=1 (α i + β i + q i + z i )+ 2 j =1 γ j =0= 2 i=1 ε i (1 + μ i )g i ( ¯ x 1 , ¯ x 2 )+ 2 i=1 λ j h j ( ¯ x 1 , ¯ x 2 ) . Kim et al. Fixed Point Theory and Applications 2011, 2011:6 http://www.fixedpointtheoryandapplications.com/content/2011/1/6 Page 10 of 13 [...]... multiobjective fractional problem J Nonlinear Convex Anal 6(2), 347–357 (2005) doi:10.1186/1687-1812-2011-6 Cite this article as: Kim et al.: On ε-optimality conditions for multiobjective fractional optimization problems Fixed Point Theory and Applications 2011 2011:6 Submit your manuscript to a journal and benefit from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on. .. GS, Lee, GM: On ε-approximate solutions for convex semidefinite optimization problems Taiwanese J Math 11(3), 765–784 (2007) Lee, GM, Lee, JH: ε-Duality theorems for convex semidefinite optimization problems with conic constraints J Inequal Appl 13 (2010) Art ID363012 Kim, GS, Lee, GM: On ε-optimality theorems for convex vector optimization problems To appear in Journal of Nonlinear and Convex Analysis... minimization problems J Optim Theory Appl 43(2), 265–276 (1984) Strodiot, JJ, Nguyen, VH, Heukemes, N: ε-Optimal solutions in nondifferentiable convex programming and some related questions Math Program 25(3), 307–328 (1983) Yokoyama, K: Epsilon approximate solutions for multiobjective programming problems J Math Anal Appl 203(1), 142–149 (1996) Yokoyama, K, Shiraishi, S: ε-Necessary conditions for convex multiobjective. .. multiplier conditions characterizing optimality without constraint qualification for convex programs SIAM J Optim 14(2), 534–547 (2003) 2 Jeyakumar, V, Wu, ZY, Lee, GM, Dinh, N: Liberating the subgradient optimality conditions from constraint qualification J Global Optim 36(1), 127–137 (2006) Kim et al Fixed Point Theory and Applications 2011, 2011:6 http://www.fixedpointtheoryandapplications.com/content/2011/1/6... of nondifferentiable nonconvex multiobjective programming J Optim Theory Appl 69(1), 153–167 (1991) Liu, JC: ε-Pareto optimality for nondifferentiable multiobjective programming via penalty function J Math Anal Appl 198(1), 248–261 (1996) Loridan, P: Necessary conditions for ε-optimality Optimality and stability in mathematical programming Math Program Stud 19, 140–152 (1982) Loridan, P: ε-Solutions... optimality for convex programs J Optim Theory Appl 93(1), 153–165 (1997) Jeyakumar, V, Lee, GM, Dinh, N: Characterizations of solution sets of convex vector minimization problems Eur J Oper Res 174(3), 1380–1395 (2006) Sawaragi, Y, Nakayama, H, Tanino, T: Theory of Multiobjective Optimization Academic Press, New York (1985) Gupta, P, Shiraishi, S, Yokoyama, K: ε-Optimality without constraint qualification for. .. MG, Mehra, A: ε-Optimality for multiobjective programming on a Banach space Eur J Oper Res 157(1), 106–112 (2004) Gutiárrez, C, Jimá, B, Novo, V: Multiplier rules and saddle-point theorems for Helbig’s approximate solutions in convex Pareto problems J Global Optim 32(3), 367–383 (2005) Hamel, A: An ε-Lagrange multiplier rule for a mathematical programming problem on Banach spaces Optimization 49(1-2),... Slater’s constraint qualification (preprint) Hiriart-Urruty, JB, Lemarechal, C: Convex Analysis and Minimization Algorithms, vols I and II Springer-Verlag, Berlin (1993) Rockafellar, RT: Convex Analysis Princeton University Press, Princeton, NJ (1970) Zalinescu, C: Convex Analysis in General Vector Space World Scientific Publishing Co Pte Ltd, Singapore (2002) Jeyakumar, V: Asymptotic dual conditions characterizing... 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) Author details 1 School of Free Major, Tongmyong University, Busan 608-711, Korea 2Department of Applied Mathematics, Pukyong National University, Busan 608-737, Korea Authors’ contributions The authors, together discussed and solved the problems in... Fixed Point Theory and Applications 2011, 2011:6 http://www.fixedpointtheoryandapplications.com/content/2011/1/6 Page 11 of 13 Thus, (iii) of Theorem 3.1 holds x ˜ (2) Let (˜ 1 , x2 ) = ( 3 , 15 ) Then (˜ 1 , x2 )is not an ε-efficient solution of (MFP) 1 , but x ˜ 2 4 (˜ 1 , x2 )is a weakly ε-efficient solution of (MFP)1 x ˜ Let C = 2 i=1 λ1 ≥0, λ2 ≥0 epi(λi hi )∗ Then C = cone co{(−1, 0, −1), (0, −1, . Slater condition) on convex opti- mization problems to obtain optimality co nditions or ε-optimality conditions for the problem. To get optimality conditions for an efficient solution of a multiobjective. Classification: 90C30, 90C46. Keywords: Weakly ε-efficient solution, ε-optimality condition, Multiobjective fractional optimization problem 1 Introduction We need constraint qualifications (for example,. RESEARC H Open Access On ε-optimality conditions for multiobjective fractional optimization problems Moon Hee Kim 1 , Gwi Soo Kim 2 and Gue Myung Lee 2* * Correspondence: gmlee@pknu.ac. kr 2 Department