# A Characterization ofE-Benson Proper Efficiency via Nonlinear Scalarization in Vector Optimization

Authors: Ke Quan Zhao; Yuan Mei Xia; Hui Guo
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

A class of vector optimization problems is considered and a characterization
ofE-Benson proper efficiency is obtained by using a nonlinear scalarization
function proposed by Göpfert et al. Some examples are given to illustrate the main
results.


---

## Body

## 1. Introduction

It is well known that approximate solutions have been playing an important role in vector optimization theory and applications. During the recent years, there are a lot of works related to vector optimization and some concepts of approximate solutions of vector optimization problems are proposed and some characterizations of these approximate solutions are studied; see, for example, [1–3] and the references therein.Recently, Chicoo et al. proposed the concept ofE-efficiency by means of improvement sets in a finite dimensional Euclidean space in [4]. E-efficiency unifies some known exact and approximate solutions of vector optimization problems. Zhao and Yang proposed a unified stability result with perturbations by virtue of improvement sets under the convergence of a sequence of sets in the sense ofWijsman in [5]. Furthermore, Gutiérrez et al. generalized the concepts of improvement sets and E-efficiency to a general Hausdorff locally convex topological linear space in [6]. Zhao et al. established linear scalarization theorem and Lagrange multiplier theorem of weak E-efficient solutions under the nearly E-subconvexlikeness in [7]. Moreover, Zhao and Yang also introduced a kind of proper efficiency, named E-Benson proper efficiency which unifies some proper efficiency and approximate proper efficiency, and obtained some characterizations of E-Benson proper efficiency in terms of linear scalarization in [8].Motivated by the works of [8, 9], by making use of a kind of nonlinear scalarization functions proposed by Göpfert et al., we establish nonlinear scalarization results of E-Benson proper efficiency in vector optimization. We also give some examples to illustrate the main results.


## 2. Preliminaries

LetX be a linear space and let Y be a real Hausdorff locally convex topological linear space. For a nonempty subset A in Y, we denote the topological interior, the topological closure, and the boundary of A by int
⁡
A, cl
⁡
A, and ∂
A, respectively. The cone generated by A is defined as
(1)
cone
⁡
A
=
⋃
α
≥
0
‍
α
A
.
A cone A
⊂
Y is pointed if A
∩
(
-
A
)
=
{
0
}. Let K be a closed convex pointed cone in Y with nonempty topological interior. For any x
,
y
∈
Y, we define
(2)
x
≤

K
y
⟺
y
-
x
∈
K
.
In this paper, we consider the following vector optimization problem:
(
VP
)
min
⁡
x
∈
D
f
(
x
)
,
where f
:
X
→
Y and ∅
≠
D
⊂
X.Definition 1 (see [4, 6]).
LetE
⊂
Y. If 0
∉
E and E
+
K
=
E, then E is said to be an improvement set with respect to K.Remark 2.
IfE
≠
∅, then, from Theorem 3.1 in [8], it is clear that int
⁡
E
≠
∅. Throughout this paper, we assume that E
≠
∅.Definition 3 (see [8]).
LetE
⊂
Y be an improvement set with respect to K. A feasible point x
0
∈
D is said to be an E-Benson proper efficient solution of (
VP
) if
(3)
cl
⁡
(
cone
⁡
(
f
(
D
)
+
E
-
f
(
x
0
)
)
)
∩
(
-
K
)
=
{
0
}
.
We denote the set of all E-Benson proper efficient solutions by x
0
∈
PAE
(
f
,
E
).
Consider the following scalar optimization problem:(
P
)
min
⁡
x
∈
Z
⁡
ϕ
(
x
)
,
where ϕ
:
X
→
ℝ, ∅
≠
Z
⊂
X. Let ϵ
≥
0 and x
0
∈
Z. If ϕ
(
x
)
≥
ϕ
(
x
0
)
-
ϵ, for all x
∈
Z, then x
0 is called an ϵ-minimal solution of (
P
). The set of all ϵ-minimal solutions is denoted by AMin
(
ϕ
,
ϵ
). Moreover, if ϕ
(
x
)
>
ϕ
(
x
0
)
-
ϵ, for all x
∈
Z, then x
0 is called a strictly ϵ-minimal solution of (
P
). The set of all strictly ϵ-minimal solutions is denoted by SAMin
(
ϕ
,
ϵ
).


## 3. A Characterization ofE-Benson Proper Efficiency

In this section, we give a characterization ofE-Benson proper efficiency of (
VP
) via a kind of nonlinear scalarization function proposed by Göpfert et al.Letξ
q
,
E
:
Y
→
ℝ
∪
{
±
∞
} be defined by
(4)
ξ
q
,
E
(
y
)
=
inf
⁡
⁡
{
s
∈
ℝ
∣
y
∈
s
q
-
E
}
,
y
∈
Y
,
with inf
⁡
∅
=
+
∞.Lemma 4.
LetE
⊂
Y be a closed improvement set with respect to K and q
∈
int
⁡
K. Then ξ
q
,
E is continuous and
(5)
{
y
∈
Y
∣
ξ
q
,
E
(
y
)
<
c
}
=
c
q
-
int
⁡
E
,
∀
c
∈
ℝ
,
{
y
∈
Y
∣
ξ
q
,
E
(
y
)
=
c
}
=
c
q
-
∂
E
,
∀
c
∈
ℝ
,
ξ
q
,
E
(
-
E
)
≤
0
,
ξ
q
,
E
(
-
∂
E
)
=
0
.Proof.
This can be easily seen from Proposition 2.3.4 and Theorem 2.3.1 in [9].
Consider the following scalar optimization problem:(
P
q
,
y
)
min
⁡
x
∈
D
ξ
q
,
E
(
f
(
x
)
-
y
)
,
where y
∈
Y, q
∈
int
⁡
K. Denote ξ
q
,
E
(
f
(
x
)
-
y
) by (
ξ
q
,
E
,
y
∘
f
)
(
x
), the set of ϵ-minimal solutions of (
P
q
,
y
) by AMin
(
ξ
q
,
E
,
y
∘
f
,
ϵ
), and the set of strictly ϵ-minimal solutions of (
P
q
,
y
) by SAMin
(
ξ
q
,
E
,
y
∘
f
,
ϵ
).Theorem 5.
LetE
⊂
Y be a closed improvement set with respect to K, q
∈
int
⁡
(
E
∩
K
) and ϵ
=
inf
⁡
{
s
∈
ℝ
+
+
∣
s
q
∈
int
⁡
(
E
∩
K
)
}. Then(i)
x
0
∈
PAE
(
f
,
E
)
⇒
x
0
∈
AMin
(
ξ
q
,
E
,
f
(
x
0
)
∘
f
,
ϵ
);
(ii)
additionally, if
cone
⁡
(
f
(
D
)
+
E
-
f
(
x
0
)
)
is a closed set, then
(6)
x
0
∈
SAMin
(
ξ
q
,
E
,
f
(
x
0
)
∘
f
,
ϵ
)
⟹
x
0
∈
PAE
(
f
,
E
)
.Proof.
We first prove (i). Assume thatx
0
∈
PAE
(
f
,
E
). Then we have
(7)
cl
⁡
(
cone
⁡
(
f
(
D
)
+
E
-
f
(
x
0
)
)
)
∩
(
-
K
)
=
{
0
}
.
Therefore,
(8)
(
f
(
D
)
+
E
-
f
(
x
0
)
)
∩
(
-
int
⁡
K
)
=
∅
.
We can prove that
(9)
(
f
(
x
0
)
-
int
⁡
E
)
∩
f
(
D
)
=
∅
.
On the contrary, there exists x
^
∈
D such that
(10)
f
(
x
^
)
-
f
(
x
0
)
∈
-
int
⁡
E
.
Hence, from Theorem 3.1 in [8], it follows that
(11)
f
(
x
^
)
-
f
(
x
0
)
∈
-
E
-
int
⁡
K
.
Therefore,
(12)
f
(
x
^
)
-
f
(
x
0
)
+
E
⊂
-
int
⁡
K
,
which contradicts (8) and so (9) holds. From Lemma 4, we obtain
(13)
{
y
∈
Y
∣
ξ
q
,
E
(
y
)
<
0
}
=
-
int
⁡
E
.
From (9), we have
(14)
(
f
(
D
)
-
f
(
x
0
)
)
∩
(
-
int
⁡
E
)
=
∅
.
By using (13) and (14), we deduce that
(15)
(
f
(
D
)
-
f
(
x
0
)
)
∩
{
y
∈
Y
∣
ξ
q
,
E
(
y
)
<
0
}
=
∅
.
Thus,
(16)
(
ξ
q
,
E
,
f
(
x
0
)
∘
f
)
(
x
)
=
ξ
q
,
E
(
f
(
x
)
-
f
(
x
0
)
)
≥
0
,
∀
x
∈
D
.
In addition, since {
s
∈
ℝ
+
+
∣
s
q
∈
int
⁡
(
E
∩
K
)
}
⊂
{
s
∈
ℝ
∣
s
q
∈
E
},
(17)
(
ξ
q
,
E
,
f
(
x
0
)
∘
f
)
(
x
0
)
=
ξ
q
,
E
(
0
)
=
inf
⁡
{
s
∈
ℝ
∣
s
q
∈
E
}
≤
ϵ
.
It follows from (16) that
(18)
(
ξ
q
,
E
,
f
(
x
0
)
∘
f
)
(
x
)
≥
(
ξ
q
,
E
,
f
(
x
0
)
∘
f
)
(
x
0
)
-
ϵ
.
Therefore, x
0
∈
AMin
(
ξ
q
,
E
,
f
(
x
0
)
∘
f
,
ϵ
).
Next, we prove (ii). Suppose thatx
0
∈
SAMin
(
ξ
q
,
E
,
f
(
x
0
)
∘
f
,
ϵ
) and x
0
∉
PAE
(
f
,
E
). Since cone
⁡
(
f
(
D
)
+
E
-
f
(
x
0
)
) is a closed set, there exist 0
≠
d
∈
-
K, λ
>
0, x
^
∈
D, and e
^
∈
E such that
(19)
d
=
λ
(
f
(
x
^
)
-
f
(
x
0
)
+
e
^
)
.
Since K is a cone,
(20)
f
(
x
^
)
-
f
(
x
0
)
+
e
^
∈
-
K
.
Therefore, we can obtain that
(21)
f
(
x
^
)
-
f
(
x
0
)
∈
-
e
^
-
K
⊂
-
E
-
K
=
-
E
.
Moreover, by Lemma 4, we have, for every c
∈
ℝ,
(22)
c
q
+
f
(
x
^
)
-
f
(
x
0
)
∈
c
q
-
E
=
c
q
-
cl
⁡
E
=
{
y
∈
Y
∣
ξ
q
,
E
(
y
)
≤
c
}
;
that is,
(23)
ξ
q
,
E
(
c
q
+
f
(
x
^
)
-
f
(
x
0
)
)
≤
c
.
Let c
=
0 in (23); then, we have
(24)
ξ
q
,
E
(
f
(
x
^
)
-
f
(
x
0
)
)
≤
0
.
On the other hand, from x
0
∈
SAMin
(
ξ
q
,
E
,
f
(
x
0
)
∘
f
,
ϵ
), it follows that
(25)
ξ
q
,
E
(
f
(
x
^
)
-
f
(
x
0
)
)
>
ξ
q
,
E
(
f
(
x
0
)
-
f
(
x
0
)
)
-
ϵ
=
ξ
q
,
E
(
0
)
-
ϵ
.
In the following, we prove
(26)
ξ
q
,
E
(
0
)
=
ϵ
.
We first point out that, for any s
≤
0, s
q
∉
E. It is obvious that 0
∉
E when s
=
0. Assume that there exists s
^
<
0 such that s
^
q
∈
E. Since q
∈
int
⁡
(
E
∩
K
)
⊂
K and -
s
^
q
∈
K, we have
(27)
0
=
s
^
q
-
s
^
q
∈
E
+
K
=
E
,
which contradicts the fact that E is an improvement set with respect to K. Hence,
(28)
ξ
q
,
E
(
0
)
=
inf
⁡
{
s
∈
ℝ
∣
0
∈
s
q
-
E
}
=
inf
⁡
{
s
∈
ℝ
+
+
∣
s
q
∈
E
}
.
Moreover, since q
∈
int
⁡
(
E
∩
K
)
⊂
K, we have, for any s
∈
ℝ
+
+, s
q
∈
K. It follows from (28) that
(29)
ξ
q
,
E
(
0
)
=
inf
⁡
{
s
∈
ℝ
+
+
∣
s
q
∈
E
∩
K
}
.
Hence (26) holds and thus, by (25), we obtain ξ
q
,
E
(
f
(
x
^
)
-
f
(
x
0
)
)
>
0, which contradicts (24) and so x
0
∈
PAE
(
f
,
E
).Remark 6.
x
0
∈
PAE
(
f
,
E
) does not imply x
0
∈
SAMin
(
ξ
q
,
E
,
f
(
x
0
)
∘
f
,
ϵ
).Example 7.
LetX
=
Y
=
ℝ
2, K
=
ℝ
+
2, f
(
x
)
=
x, and
(30)
E
=
{
(
x
1
,
x
2
)
∣
x
1
+
x
2
≥
1
,
x
1
≥
0
,
x
2
≥
0
}
,
D
=
{
(
x
1
,
x
2
)
∣
x
1
-
x
2
=
0
,
-
1
2
≤
x
1
≤
0
}
.
Clearly, K is a closed convex cone and E is a closed improvement set with respect to K. Let x
0
=
(
0,0
)
∈
D and q
=
(
1,1
)
∈
int
⁡
(
E
∩
K
). Then ϵ
=
1
/
2 since
(31)
cl
⁡
(
cone
⁡
(
f
(
D
)
+
E
-
f
(
x
0
)
)
)
∩
(
-
K
)
=
{
(
x
1
,
x
2
)
∣
x
1
+
x
2
≥
0
}
∩
(
-
ℝ
+
2
)
=
{
(
0,0
)
}
.
Hence
(32)
x
0
∈
PAE
(
f
,
E
)
.
For any x
∈
D,
(33)
ξ
q
,
E
(
f
(
x
)
-
f
(
x
0
)
)
=
ξ
q
,
E
(
f
(
x
)
)
=
inf
⁡
{
s
∈
ℝ
∣
f
(
x
)
∈
s
q
-
E
}
≥
0
=
1
2
-
1
2
=
ξ
q
,
E
(
0
)
-
ϵ
.
Therefore,
(34)
x
0
∈
AMin
(
ξ
q
,
E
,
f
(
x
0
)
∘
f
,
ϵ
)
.
However, there exists x
^
=
(
-
1
/
2
,
-
1
/
2
)
∈
D such that
(35)
ξ
q
,
E
(
f
(
x
^
)
-
f
(
x
0
)
)
=
ξ
q
,
E
(
f
(
x
^
)
)
=
inf
⁡
{
s
∈
ℝ
∣
f
(
x
^
)
∈
s
q
-
E
}
=
0
=
1
2
-
1
2
=
ξ
q
,
E
(
0
)
-
ϵ
.
Hence
(36)
x
0
∉
SAMin
(
ξ
q
,
E
,
f
(
x
0
)
∘
f
,
ϵ
)
.Remark 8.
Theorem5(ii) may not be true if the closedness of cone
⁡
(
f
(
D
)
+
E
-
f
(
x
0
)
) is removed and the following example can illustrate it.Example 9.
LetX
=
Y
=
ℝ
2, K
=
ℝ
+
2, f
(
x
)
=
x, and
(37)
E
=
{
(
x
1
,
x
2
)
∣
x
1
+
x
2
≥
1
,
x
1
≥
0
,
x
2
≥
1
2
}
,
D
=
{
(
x
1
,
x
2
)
∣
x
1
≤
0
,
x
2
=
0
}
.
Clearly, K is a closed convex cone and E is a closed improvement set with respect to K. Let x
0
=
(
0,0
)
∈
D and q
=
(
1,1
)
∈
int
⁡
(
E
∩
K
). Then ϵ
=
1
/
2 and
(38)
cone
⁡
(
f
(
D
)
+
E
-
f
(
x
0
)
)
=
{
(
x
1
,
x
2
)
∣
x
1
∈
ℝ
,
x
2
>
0
}
∪
{
(
0,0
)
}
is not a closed set, since for any x
∈
D
(39)
ξ
q
,
E
(
f
(
x
)
-
f
(
x
0
)
)
=
ξ
q
,
E
(
f
(
x
)
)
=
inf
⁡
{
s
∈
ℝ
∣
f
(
x
)
∈
s
q
-
E
}
=
1
2
>
1
2
-
1
2
=
ξ
q
,
E
(
0
)
-
ϵ
.
Therefore,
(40)
x
0
∈
SAMin
(
ξ
q
,
E
,
f
(
x
0
)
∘
f
,
ϵ
)
.
However,
(41)
cl
⁡
(
cone
⁡
(
f
(
D
)
+
E
-
f
(
x
0
)
)
)
∩
(
-
K
)
=
{
(
x
1
,
x
2
)
∣
x
1
∈
ℝ
,
x
2
≥
0
}
∩
(
-
ℝ
+
2
)
=
{
(
x
1
,
x
2
)
∣
x
1
≤
0
,
x
2
=
0
}
≠
{
(
0,0
)
}
.
Therefore,
(42)
x
0
∉
PAE
(
f
,
E
)
.Remark 10.
Theorem5(ii) may not be true if x
0
∈
SAMin
(
ξ
q
,
E
,
f
(
x
0
)
∘
f
,
ϵ
) is replaced by x
0
∈
AMin
(
ξ
q
,
E
,
f
(
x
0
)
∘
f
,
ϵ
) and the following example can illustrate it.Example 11.
LetX
=
Y
=
ℝ
2, K
=
ℝ
+
2, f
(
x
)
=
x, and
(43)
E
=
{
(
x
1
,
x
2
)
∣
x
1
+
x
2
≥
1
,
x
1
≥
1
2
,
x
2
≥
0
}
∪
{
(
x
1
,
x
2
)
∣
x
1
≤
1
2
,
x
2
≥
1
2
}
,
D
=
{
(
x
1
,
x
2
)
∣
x
1
-
x
2
=
0
,
-
1
2
≤
x
1
≤
0
}
.
Clearly, K is a closed convex cone and E is a closed improvement set with respect to K. Let x
0
=
(
0,0
)
∈
D and q
=
(
1,1
)
∈
int
⁡
(
E
∩
K
). Then ϵ
=
1
/
2 and
(44)
cone
⁡
(
f
(
D
)
+
E
-
f
(
x
0
)
)
=
{
(
x
1
,
x
2
)
∣
x
1
∈
ℝ
,
x
2
≥
0
}
∪
{
(
x
1
,
x
2
)
∣
x
1
+
x
2
≥
0
,
x
1
≥
0
,
x
2
≤
0
}
is a closed set, since for any x
∈
D
(45)
ξ
q
,
E
(
f
(
x
)
-
f
(
x
0
)
)
=
ξ
q
,
E
(
f
(
x
)
)
=
inf
⁡
{
s
∈
ℝ
∣
f
(
x
)
∈
s
q
-
E
}
≥
0
=
1
2
-
1
2
=
ξ
q
,
E
(
0
)
-
ϵ
.
Therefore,
(46)
x
0
∈
AMin
(
ξ
q
,
E
,
f
(
x
0
)
∘
f
,
ϵ
)
.
However, there exists x
^
=
(
-
1
/
2
,
-
1
/
2
)
∈
D such that
(47)
ξ
q
,
E
(
f
(
x
^
)
-
f
(
x
0
)
)
=
ξ
q
,
E
(
f
(
x
^
)
)
=
inf
⁡
{
s
∈
ℝ
∣
f
(
x
^
)
∈
s
q
-
E
}
=
0
=
1
2
-
1
2
=
ξ
q
,
E
(
0
)
-
ϵ
.
Hence,
(48)
x
0
∉
SAMin
(
ξ
q
,
E
,
f
(
x
0
)
∘
f
,
ϵ
)
.
Moreover,
(49)
cl
⁡
(
cone
⁡
(
f
(
D
)
+
E
-
f
(
x
0
)
)
)
∩
(
-
K
)
=
{
(
x
1
,
x
2
)
∣
x
1
∈
ℝ
,
x
2
≥
0
}
∪
{
(
x
1
,
x
2
)
∣
x
1
+
x
2
≥
0
,
x
1
≥
0
,
x
2
≤
0
}
∩
(
-
ℝ
+
2
)
=
{
(
x
1
,
x
2
)
∣
x
1
≤
0
,
x
2
=
0
}
≠
{
(
0,0
)
}
.
Therefore,
(50)
x
0
∉
PAE
(
f
,
E
)
.



---

*Source: 649756-2014-04-28.xml*