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.