# A Kind of Unified Proper Efficiency in Vector Optimization

Authors: Ke Quan Zhao; Yuan Mei Xia
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Based on the ideas of the classical Benson proper efficiency, a new kind of unified proper efficiency namedS-Benson proper efficiency is introduced by using Assumption (B) proposed by Flores-Bazán and Hernández, which unifies some known exact and approximate proper efficiency including (
C
,
ε
)-proper efficiency and E-Benson proper efficiency in vector optimization. Furthermore, a characterization of S-Benson proper efficiency is established via a kind of nonlinear scalarization functions introduced by Göpfert et al.


---

## Body

## 1. Introduction

It is well known that approximate solutions have been playing an important role in vector optimization. Since Kutateladze initially introduced the concept of approximate solutions in [1], a lot of research achievements of approximate solutions have been obtained for vector optimization problems. Loridan proposed ε-efficient solutions of vector optimization problems and gave some properties in [2]. In a general topological vector space, Rong and Wu proposed ε-weak efficient solutions of vector optimization problems with set-valued maps and obtained some linear scalarization theorems, Lagrangian multipliers theorems, saddle point theorems, and duality theorems in [3]. Recently, Gutiérrez et al. introduced the concept of coradiant set and proposed (
C
,
ε
)-efficient solutions which extend and unify some known different notions of approximate solutions in [4]. Gao et al. proposed the concept of properly approximate efficient solutions by means of coradiant set and established some linear and nonlinear scalarization results in [5]. Furthermore, Gutiérrez et al. obtained some characterizations of this kind of approximate solutions in terms of linear scalarization in [6].Moreover, Debreu introduced the concept of free disposal sets to deal with mathematical economic problems in [7]. In a finite dimensional space, Chicco et al. introduced the concepts of improvement sets and E-efficient solutions and obtained some characterizations in [8]. Improvement sets are close to free disposal sets and can be applied to study vector optimization problems as an important tool. In particular, Zhao and Yang obtained a unified stability result with perturbations by means of improvement sets in [9]. Furthermore, Gutiérrez et al. generalized the concepts of improvement sets and E-efficient solutions to a general real locally convex Hausdorff topological vector space and studied some linear scalarization results in [10]. Zhao and Yang proposed E-weak efficient solutions of vector optimization problems with set-valued maps and established some linear scalarization theorems, Lagrange multiplier theorems, saddle point criteria, and duality in [11]. Zhao and Yang introduced the concept of E-Benson proper efficiency which unifies some proper efficiency and obtained some linear scalarization theorems and Lagrange multiplier theorems of this kind of proper efficiency in [12]. Flores-Bazán and Hernández proposed Assumption (B) and obtained some complete scalarizations of solution sets of a class of unified vector optimization problems via nonlinear scalarization in [13]. In addition, Flores-Bazán and Hernández obtained some optimality conditions of a class of unified vector optimization problems under Assumption (B) in [14].Motivated by the works of [4, 5, 12, 13], we present a new kind of unified proper efficiency named S-Benson proper efficiency by using Assumption (B) proposed by Flores-Bazán and Hernández. This kind of proper efficiency unifies some known exact and approximate proper efficiency including (
C
,
ε
)-proper efficiency and E-Benson proper efficiency in vector optimization. Furthermore, we also give a characterization of S-Benson proper efficiency via nonlinear scalarization.


## 2. Preliminaries

LetX be a linear space and Y a real Hausdorff locally convex topological linear space. For a subset A of Y, we denote the topological interior, the topological closure, the boundary, and the complement of A by int
⁡
A, cl
⁡
A, ∂
A, and Y
∖
A, respectively. A set A is solid if int
⁡
A
≠
∅ and is proper if A is nonempty and A
≠
Y. The cone generated by A is defined as
(1)
cone
⁡
A
=
{
α
a
∣
α
≥
0
,
a
∈
A
}
.
Let Y
* denote the topological dual space of Y. The positive dual cone of a subset A
⊂
Y is defined as
(2)
A
+
=
{
y
*
∈
Y
*
∣
〈
y
*
,
y
〉
≥
0
,
∀
y
∈
A
}
.
Let K be a closed convex pointed cone in Y with nonempty topological interior. For any x
,
y
∈
Y, we define
(3)
x
≤
K
y
⟺
y
-
x
∈
K
.
In this paper, we consider the following vector optimization problem:
(
VP
)
min
⁡
f
(
x
)
s
.
t
.
x
∈
D
,
where f
:
X
→
Y and ∅
≠
D
⊂
X.We say thatA is a coradiant set if A satisfies α
d
∈
A for every d
∈
A, α
>
1. Let C
⊂
Y be a proper solid coradiant set and define
(4)
C
(
ε
)
=
ε
C
,
∀
ε
>
0
,
C
(
0
)
=
⋃
ε
>
0
C
(
ε
)
.Lemma 1 (see [5]).
LetC be a proper solid convex coradiant set. Then,(i)
C
(
0
)
+
C
(
ε
)
⊂
C
(
ε
)
,
∀
ε
≥
0;
(ii)
int
⁡
(
cl
⁡
C
(
ε
)
)
=
int
⁡
C
(
ε
)
,
∀
ε
>
0.Definition 2 (see [5]).
Letε
≥
0. A feasible point x
¯
∈
D is said to be a (
C
,
ε
)-proper efficient solution of (
VP
) if
(5)
cl
⁡
cone
⁡
(
f
(
D
)
+
C
(
ε
)
-
f
(
x
¯
)
)
∩
(
-
C
(
0
)
)
⊂
{
0
}
.Definition 3 (see [10]).
A nonempty setE
⊂
Y is said to be an improvement set with respect to K if 0
∉
E and E
+
K
=
E.Lemma 4 (see [10]).
LetE
⊂
Y be a nonempty set. If E is an improvement set with respect to K, then E
+
⊂
K
+. Additionally, if E
⊂
K, then E
+
=
K
+.Definition 5 (see [12]).
LetE
⊂
Y be an improvement set with respect to K. A feasible point x
¯
∈
D is said to be an E-Benson proper efficient solution of (
VP
) if
(6)
cl
⁡
(
cone
⁡
(
f
(
D
)
+
E
-
f
(
x
¯
)
)
)
∩
(
-
K
)
=
{
0
}
.
Flores-Bazán and Hernández introduced Assumption B as follows.Assumption B (see [13]). Consider that 0
≠
q
∈
Y and S
⊂
Y is a proper (not necessary closed) set such that 0
∈
∂
S and cl
⁡
(
Y
∖
(
-
S
)
)
+
ℝ
+
+
q
⊂
int
⁡
(
Y
∖
(
-
S
)
).Remark 6.
From Assumption B, we have the equivalence(7)
cl
⁡
(
Y
∖
(
-
S
)
)
+
ℝ
+
+
q
⊂
int
⁡
(
Y
∖
(
-
S
)
)
⟺
cl
⁡
S
+
ℝ
+
+
q
⊂
int
⁡
S
.Lemma 7 (see [15]).
LetS
⊂
Y be any nonempty subset. Then, cl
⁡
(
cone
⁡
S
)
=
cl
⁡
(
cone
⁡
(
cl
⁡
S
)
).


## 3. A Kind of Unified Proper Efficiency

In this section, we propose a kind of unified proper efficiency of(
VP
) by means of Assumption B by using the idea of the classical Benson proper efficiency and discuss some relations with other proper efficiency.Definition 8.
Letq and S satisfy Assumption B. One says that x
¯
∈
D is a S-Benson proper efficient solution of (
VP
) if
(8)
cl
⁡
(
cone
⁡
(
f
(
D
)

+

S
-
f
(
x
¯
)
)
)
∩
(
-
cl
⁡
(
cone
⁡
(
conv
(
S
)
)
)
)

=

{
0
}
.
Denote by PAE
(
f
,
S
) the set of S-Benson proper efficient solutions of (
VP
).Example 9.
LetY
=
ℝ
2, q
=
(
0,1
), f
(
x
)
=
x, D
=
{
(
x
1
,
x
2
)
∣
x
1
≤
0
,
x
2
≥
0
}, and
(9)
S
=
{
(
x
1
,
x
2
)
∣
x
1
-
x
2
≤
0
,
x
1
∈
ℝ
,
x
2
>
0
}
∪
{
(
x
1
,
x
2
)
∣
-
1
≤
x
1
≤
0
,
x
2
=
0
}
.
Since
(10)
cl
⁡
S

+

ℝ
+
+
q
=
{
(
x
1
,
x
2
)
∣
x
1

-

x
2
<
0
,
x
1
∈
ℝ
,
x
2
>
0
}
=
int
⁡
S
,
then, from Remark 6, it follows that q and S satisfy Assumption B. Let x
¯
=
(
0,0
)
∈
D. Since
(11)
cl
⁡
(
cone
⁡
(
f
(
D
)
+
S
-
f
(
x
¯
)
)
)
=
cl
⁡
(
cone
⁡
(
conv
(
S
)
)
)
=
{
(
x
1
,
x
2
)
∣
x
1
-
x
2
≤
0
,
x
1
∈
ℝ
,
x
2
≥
0
}
,
then
(12)
cl
⁡
(
cone
⁡
(
f
(
D
)

+

S

-

f
(
x
¯
)
)
)
∩
(
-
cl
⁡
(
cone
⁡
(
conv
(
S
)
)
)
)
=
{
0
}
.
Therefore, x
¯
∈
PAE
(
f
,
S
).In the following, we discuss some relations betweenS-Benson proper efficiency and some other proper efficiency.Theorem 10.
LetK
⊂
Y be a pointed closed convex cone, S
=
int
⁡
K, and q
∈
int
⁡
K. Then, S-Benson proper efficiency reduces to the Benson proper efficiency.Proof.
SinceK is a convex cone, then we have int
⁡
K
+
K
=
int
⁡
K and hence, by 0
∉
S, we can obtain that S is an improvement set with respect to K. Then, it follows from Remark 3.2 in [12] that
(13)
cl
⁡
(
Y
∖
(
-
S
)
)
+
ℝ
+
+
q
⊂
int
⁡
(
Y
∖
(
-
S
)
)
.
For 0
∈
∂
S, q and S satisfy Assumption B. Assume that x
¯ is a S-Benson proper efficient solution of (
VP
) and then, from Proposition 4.1 in [16], we have
(14)
cl
⁡
(
cone
⁡
(
f
(
D
)
+
K
-
f
(
x
¯
)
)
)
∩
(
-
K
)
=
cl
⁡
(
cone
⁡
(
f
(
D
)
+
int
⁡
K
-
f
(
x
¯
)
)
)
∩
(
-
K
)
=
cl
⁡
(
cone
⁡
(
f
(
D
)
+
S
-
f
(
x
¯
)
)
)
∩
(
-
cl
⁡
(
cone
⁡
(
conv
(
S
)
)
)
)
=
{
0
}
,
which implies that x
¯ is a Benson proper efficient solution of (
VP
).Theorem 11.
LetK
⊂
Y be a pointed closed convex set and q
∈
int
⁡
K. If S
=
E
⊂
K is an improvement set with respect to K and 0
∈
∂
S, then S-Benson proper efficiency reduces to the E-Benson proper efficiency.Proof.
From Remark 3.2 in [12], we know that q and S satisfy Assumption B. Assume that x
¯ is S-Benson proper efficient solution of (
VP
). We first point out that
(15)
cl
⁡
(
cone
⁡
(
conv
S
)
)
=
K
.
In fact, since S
⊂
K, then we only need to prove
(16)
K
⊂
cl
⁡
(
cone
⁡
(
conv
S
)
)
.
Suppose that there exists k
0
∈
K such that k
0
∉
cl
⁡
(
cone
⁡
(
conv
S
)
). By applying separation theorem for convex sets, it follows that there exists λ
∈
Y
*
∖
{
0
Y
*
} such that
(17)
〈
λ
,
k
0
〉
>
〈
λ
,
e
〉
,
∀
e
∈
cl
⁡
(
cone
⁡
(
conv
S
)
)
.
Let e
=
0; we have
(18)
〈
λ
,
k
0
〉
>
0
.
Furthermore, we can show that -
λ
∈
(
cl
⁡
(
cone
⁡
(
conv
S
)
)
)
+
=
S
+. Since S is an improvement set with respect to K and by Lemma 4, we can obtain
(19)
-
λ
∈
E
+
=
K
+
,
which implies 〈
λ
,
k
0
〉
≤
0. This contradicts (18) and then (15) holds. Hence,
(20)
cl
⁡
(
cone
⁡
(
f
(
D
)
+
E
-
f
(
x
¯
)
)
)
∩
(
-
K
)
=
cl
⁡
(
cone
⁡
(
f
(
D
)
+
S
-
f
(
x
¯
)
)
)
∩
(
-
cl
⁡
(
cone
⁡
(
conv
S
)
)
)
=
{
0
}
.
This means that x
¯ is an E-Benson proper efficient solution of (
VP
).Theorem 12.
LetC be a proper solid convex coradiant set, q
∈
int
⁡
C
(
0
), ε
≥
0, S
=
C
(
ε
), and 0
∈
∂
S. Then, S-Benson proper efficiency reduces to (
C
,
ε
)-proper efficiency.Proof.
From the convexity ofS and Lemma 1(i), we have
(21)
cl
⁡
S
+
cl
⁡
C
(
0
)
=
cl
⁡
C
(
ε
)

+

cl
⁡
C
(
0
)
⊂
cl
⁡
(
C
(
ε
)

+

C
(
0
)
)
⊂
cl
⁡
C
(
ε
)
=
cl
⁡
S
,
and so, from 0
∈
cl
⁡
C
(
0
), it follows that
(22)
cl
⁡
S
+
cl
⁡
C
(
0
)
=
cl
⁡
S
.
We first point out that q and S satisfy Assumption B. In fact, we only need to prove
(23)
Y
∖
(
-
int
⁡
S
)
+
ℝ
+
+
q
⊂
Y
∖
(
-
cl
⁡
S
)
.
For any x
∈
Y
∖
(
-
int
⁡
S
)
+
ℝ
+
+
q, we only need to prove x
∉
-
cl
⁡
S. On the contrary, suppose that -
x
∈
cl
⁡
S. Since x
∈
Y
∖
(
-
int
⁡
S
)
+
ℝ
+
+
q, then there exist
(24)
x
1
∈
Y
∖
(
-
int
⁡
S
)
,
x
2
∈
ℝ
+
+
q
such that x
=
x
1
+
x
2; that is, -
x
1
=
-
x
+
x
2. Hence, from Lemma 1(ii) and (22), we have
(25)
-
x
1
∈
cl
⁡
S
+
ℝ
+
+
q
⊂
cl
⁡
S
+
int
⁡
C
(
0
)
⊂
int
⁡
(
cl
⁡
S
+
C
(
0
)
)
⊂
int
⁡
(
cl
⁡
S
+
cl
⁡
C
(
0
)
)
=
int
⁡
(
cl
⁡
S
)
=
int
⁡
S
,
which contradicts x
1
∈
Y
∖
(
-
int
⁡
S
) and so q and S satisfy Assumption B. Furthermore, from S
⊂
C
(
0
) and by means of (22), similar with the proof of (15), we have
(26)
cl
⁡
(
cone
⁡
(
conv
(
cl
⁡
S
)
)
)
=
cl
⁡
C
(
0
)
.
From Lemma 7, it follows that
(27)
cl
⁡
(
cone
⁡
(
conv
S
)
)
=
cl
⁡
(
cone
⁡
S
)
=
cl
⁡
(
cone
⁡
(
cl
⁡
S
)
)
=
cl
⁡
(
cone
⁡
(
conv
(
cl
⁡
S
)
)
)
=
cl
⁡
C
(
0
)
.
If x
¯ is S-Benson proper efficient solution of (
VP
), then
(28)
cl
⁡
(
cone
⁡
(
f
(
D
)
+
C
(
ε
)
-
f
(
x
¯
)
)
)
∩
(
-
cl
⁡
C
(
0
)
)
=
cl
⁡
(
cone
⁡
(
f
(
D
)
+
S
-
f
(
x
¯
)
)
)
∩
(
-
cl
⁡
(
cone
⁡
(
conv
S
)
)
)
=
{
0
}
.
It follows that
(29)
cl
⁡
(
cone
⁡
(
f
(
D
)
+
C
(
ε
)
-
f
(
x
¯
)
)
)
∩
(
-
C
(
0
)
)
⊂
{
0
}
,
which implies that x
¯ is a (
C
,
ε
)-proper efficient solution of (
VP
).


## 4. A Characterization via Nonlinear Scalarization

In this section, we give a characterization ofS-Benson proper efficiency of (
VP
) via a kind of nonlinear scalarization function proposed by Göpfert et al.Definition 13.
Letξ
q
,
S
:
Y
→
ℝ
∪
{
±
∞
} be defined by
(30)
ξ
q
,
S
(
y
)
=
inf
⁡
{
t
∈
ℝ
∣
y
∈
t
q
-
S
}
,
y
∈
Y
,
with inf
⁡
∅
=
+
∞.Flores-Bazán and Hernández obtained the following nonconvex separation theorem.Lemma 14 (see [13]).
Letq and S satisfy Assumption B. Then,
(31)
{
y
∈
Y
∣
ξ
q
,
S
(
y
)
<
c
}
=
c
q
-
int
⁡
S
,
∀
c
∈
ℝ
,
{
y
∈
Y
∣
ξ
q
,
S
(
y
)
=
c
}
=
c
q
-
∂
S
,
∀
c
∈
ℝ
,
ξ
q
,
S
(
-
S
)
≤
0
,
ξ
q
,
S
(
-
∂
S
)
=
0
.
We consider the following scalar optimization problem
(32)
(
P
q
,
y
)
min
⁡
x
∈
D
ξ
q
,
S
(
f
(
x
)
-
y
)
,
where y
∈
Y and q
∈
Y. Denote ξ
q
,
S
(
f
(
x
)
-
y
) by (
ξ
q
,
S
,
y
∘
f
)
(
x
). Let ϵ
≥
0 and x
¯
∈
D. If
(33)
(
ξ
q
,
S
,
y
∘
f
)
(
x
)
≥
(
ξ
q
,
S
,
y
∘
f
)
(
x
¯
)
-
ϵ
,
∀
x
∈
D
,
then x
¯ is called an ϵ-minimal solution of (
P
q
,
y
). Denote the set of ϵ-minimal solutions of (
P
q
,
y
) by A
M
i
n
(
ξ
q
,
S
,
y
∘
f
,
ϵ
).Theorem 15.
Letq
∈
int
⁡
S and S satisfy Assumption B and β
=
inf
⁡
{
t
∈
ℝ
+
∣
t
q
∈
S
}. Then,
(34)
x
¯
∈
P
A
E
(
f
,
S
)
⟹
x
¯
∈
A
M
i
n
(
ξ
q
,
S
,
f
(
x
¯
)
∘
f
,
β
)
.Proof.
Sincex
¯
∈
PAE
(
f
,
S
), then
(35)
cl
⁡
(
cone
⁡
(
f
(
D
)
+
S
-
f
(
x
¯
)
)
)
∩
(
-
cl
⁡
(
cone
⁡
(
conv
S
)
)
)
=
{
0
}
,
and it follows that
(36)
(
f
(
D
)
+
S
-
f
(
x
¯
)
)
∩
(
-
int
⁡
(
cl
⁡
(
cone
⁡
(
conv
S
)
)
)
)
=
∅
.
Therefore,
(37)
(
f
(
x
¯
)
-
(
S
+
int
⁡
(
cl
⁡
(
cone
⁡
(
conv
S
)
)
)
)
)
∩
f
(
D
)
=
∅
.
Furthermore, we can verify that
(38)
int
⁡
S
⊂
S
+
int
⁡
(
cl
⁡
(
cone
⁡
(
conv
S
)
)
)
.
In fact, from Lemma 2.5 in [17], we have
(39)
int
⁡
S
⊂
int
⁡
(
S
+
cl
⁡
(
cone
⁡
(
conv
S
)
)
)
⊂
int
⁡
(
cl
⁡
S
+
cl
⁡
(
cone
⁡
(
conv
S
)
)
)
⊂
int
⁡
(
cl
⁡
(
S
+
cone
⁡
(
conv
S
)
)
)
=
S
+
int
⁡
(
cone
⁡
(
conv
S
)
)
=
S
+
int
⁡
(
cl
⁡
(
cone
⁡
(
conv
S
)
)
)
.
Hence, from (37), we deduce that
(40)
(
f
(
D
)
-
f
(
x
¯
)
)
∩
(
-
int
⁡
S
)
=
∅
.
From Lemma 14, we can obtain that, for all c
∈
ℝ,
(41)
{
y
∈
Y
∣
ξ
q
,
S
(
y
)
<
c
}
=
c
q
-
int
⁡
S
.
Let c
=
0 in (41); then we have
(42)
{
y
∈
Y
∣
ξ
q
,
S
(
y
)
<
0
}
=
-
int
⁡
S
.
It follows from (40) that
(43)
(
f
(
D
)
-
f
(
x
¯
)
)
∩
{
y
∈
Y
∣
ξ
q
,
S
(
y
)
<
0
}
=
∅
.
Therefore,
(44)
(
ξ
q
,
S
,
f
(
x
¯
)
∘
f
)
(
x
)
=
ξ
q
,
S
(
f
(
x
)
-
f
(
x
¯
)
)
≥
0
,
∀
x
∈
D
.
Now, we calculate (
ξ
q
,
S
,
f
(
x
¯
)
∘
f
)
(
x
¯
). In fact,
(45)
(
ξ
q
,
S
,
f
(
x
¯
)
∘
f
)
(
x
¯
)
=
ξ
q
,
S
(
f
(
x
¯
)
-
f
(
x
¯
)
)
=
ξ
q
,
S
(
0
)
=
inf
⁡
{
t
∈
ℝ
∣
0
∈
t
q
-
S
}
=
inf
⁡
{
t
∈
ℝ
∣
t
q
∈
S
}
≤
inf
⁡
{
t
∈
ℝ
+
∣
t
q
∈
S
}
=
β
.
Hence, from (44), we have
(46)
(
ξ
q
,
S
,
f
(
x
¯
)
∘
f
)
(
x
)
≥
(
ξ
q
,
S
,
f
(
x
¯
)
∘
f
)
(
x
¯
)
-
β
,
∀
x
∈
D
,
which means that
(47)
x
¯
∈
AMin
(
ξ
q
,
S
,
f
(
x
¯
)
∘
f
,
β
)
.



---

*Source: 636907-2014-04-30.xml*