# Sufficiency and Duality for Multiobjective Programming under New Invexity

Authors: Yingchun Zheng; Xiaoyan Gao
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

A class of multiobjective programming problems including inequality constraints is considered. To this aim, some new concepts of generalizedF
,
P-type I and F
,
P-type II functions are introduced in the differentiable assumption by using the sublinear function F. These new functions are used to establish and prove the sufficient optimality conditions for weak efficiency or efficiency of the multiobjective programming problems. Moreover, two kinds of dual models are formulated. The weak dual, strong dual, and strict converse dual results are obtained under the aforesaid functions.


---

## Body

## 1. Introduction

The field of multiobjective programming, also called vector programming, has grown remarkably in different directions since the 1980s. Many researchers have been interested in the optimality conditions and duality results for the weak efficient solution and efficient solution of the multiobjective programming problems. A large literature was developed around the sufficiency and duality in multiobjective optimization [1]. In [2], Jayswal obtained the Kuhn-Tucker type sufficient optimality conditions for a feasible solution to be an efficient solution and the Mond-Weir type duality results are also presented. More specifically, Gao [3] considered the nonsmooth multiobjective semi-infinite programming and obtained several sufficient conditions and duality results. Also, Bae et al. [4] established duality theorems for nondifferentiable multiobjective programming problems under generalized convexity assumptions. Recently, Kim and Lee [5] introduced the nonsmooth multiobjective programming problems involving locally Lipschitz functions and support functions. For more descriptions of the multiobjective programming, we refer to [6–9].Furthermore, various types of generalizations of convexity theory have played an important role in the evolution of the multiobjective programming. During the past decades, the generalizations of invexity were enriched with and without differentiability assumptions. For example, we can see [10–12]. In particular, Nahak and Mohapatra [13] introduced the concept of ρ-η
,
θ-invexity function and discussed a class of multiobjective programming problems by using the new generalized functions. Padhan and Nahak [14] introduced higher-order ρ-η
,
θ-invexity functions for studying two different pairs of higher-order symmetric dual programs. In [15], Antczak extended the concept of ϕ
,
ρ-invexity for differentiable optimization problems to the case of mathematical programming problems with locally Lipschitz functions. In [16], Antczak and Stasiak introduced the concept of ϕ
,
ρ-invexity for strong compact Lipschitz mappings in Banach spaces. Sufficient optimality conditions and Mond-Weir duality theorems are derived by the assumption of generalized nonsmooth (
ϕ
,
ρ
)-invexity between Banach spaces. In [17], based upon the F-convexity and ρ-convexity, the authors defined the (
ϕ
,
ρ
)-V-type I functions to consider a class of nonsmooth multiobjective programming problems. The invexity of functions is more useful in the research of optimization.In this paper, we consider the multiobjective programming problems. The new class of generalized invexity functions, namely, pseudoinvex(
F
,
P
)
-type I (pseudoinvex (
F
,
P
)-type II, etc.) are introduced. The sufficient optimality conditions are obtained. Then weak, strong, and strict converse dual results are also established for two types of dual models related to multiobjective programming problems involving the new generalized invex functions.


## 2. Notations and Preliminaries

Throughout the paper, we use the following conventions for vectors inR
n: (1)
x
≦
y
if and only
x
i
≤
y
i
,
∀
i
=
1,2
,
…
,
n
;
x
≤
y
if and only
x
i
≦
y
i
,
∀
i
=
1,2
,
…
,
n
,
x
≠
y
;
x
<
y
if and only
x
i
<
y
i
,
∀
i
=
1,2
,
…
,
n
.In this paper, we consider the following multiobjective programming problem:M
P
Minimize
f
x
=
f
1
x
,
f
2
x
,
…
,
f
k
x
subject to
g
j
x
≦
0
,
j
=
1,2
,
…
,
m
,
x
∈
X
,where X
⊆
R
n is an open set and f
i
:
X
→
R
,
i
∈
K
=
{
1,2
,
…
,
k
} and g
j
:
X
→
R
,
j
∈
M
=
{
1,2
,
…
,
m
} are differentiable on X. Let X
0
=
x
∣
g
j
(
x
)
≦
0
,
j
∈
M be the set of all feasible solutions of M
P.Definition 1.
A feasible solutionx
-
∈
X
0 of M
P is said to be a weakly efficient solution for M
P, if there exists no other x
∈
X
0, such that (2)
f
x
<
f
x
-
.Definition 2.
A feasible solutionx
-
∈
X
0 of M
P is said to be an efficient solution for M
P, if there exists no other x
∈
X
0, such that (3)
f
x
≤
f
x
-
.Definition 3.
A functionF
:
X
×
X
×
R
n
→
R is sublinear if, for any x
,
x
-
∈
X, (4)
F
x
,
x
-
;
a
1
+
a
2
≦
F
x
,
x
-
;
a
1
+
F
x
,
x
-
;
a
2
,
∀
a
1
,
a
2
∈
R
n
,
F
x
,
x
-
;
l
a
=
l
F
x
,
x
-
;
a
,
∀
l
∈
R
,
l
≧
0
,
a
∈
R
n
.Remark 4.
It should be noted thatF
(
x
,
x
-
;
0
)
=
0.
Letf and g be differentiable at a given point x
-
∈
X
,
η
:
X
×
X
→
R
n,  α
i
,
β
j
:
X
×
X
→
R
+
∖
0,  d
:
X
×
X
→
R,  ρ
i
,
τ
j
∈
R
,
b
:
X
×
X
→
R
+
∖
{
0
},  P
∈
R
n
,
i
∈
K
,
j
∈
M.Definition 5.
(
f
,
g
) is said to be pseudoinvex (
F
,
P
)-type I at x
-
∈
X, if there exists functions b
,
α
i
,
β
j
,
η
,
d and ρ
i
,
τ
j
∈
R
,
P
∈
R
n, such that each x
∈
X; the following inequalities hold:(5)
b
x
,
x
-
f
x
-
f
x
-
<
0
⟹
F
x
,
x
-
;
α
i
x
,
x
-
η
T
x
,
x
-
∇
f
i
x
-
P
+
ρ
i
d
2
x
,
x
-
<
0
,
i
∈
K
,
F
x
,
x
-
;
β
j
x
,
x
-
η
T
x
,
x
-
∇
g
j
x
-
P
+
τ
j
d
2
x
,
x
-
<
0
,
j
∈
M
.Example 6.
LetX
=
-
1,1
⊂
R. Let the functions f
:
X
→
R, g
:
X
→
R, and b
:
X
×
X
→
R
+
∖
{
0
} be defined by(6)
f
x
=
e
x
-
3
x
,
g
x
=
-
x
3
-
2
x
-
5
,
b
x
,
x
-
=
x
2
+
x
-
2
+
1
,and the functions F
:
X
×
X
×
R
→
R, η
:
X
×
X
→
R be given by (7)
F
x
,
x
-
;
a
=
a
2
-
x
-
2
,
η
x
,
x
-
=
2
x
-
x
-
.Moreover, the functions α
:
X
×
X
→
R
+
∖
{
0
},  β
:
X
×
X
→
R
+
∖
0, and d
:
X
×
X
→
R are defined by (8)
α
x
,
x
-
=
x
-
2
+
1
,
β
x
,
x
-
=
x
-
+
2
,
d
x
,
x
-
=
x
-
x
-
.Then, let (9)
ρ
=
-
1
,
τ
=
-
2
,
P
=
1
.Now, we have(10)
b
x
,
x
-
f
x
-
f
x
-
<
0
,
where
x
-
<
x
.Then(11)
F
x
,
x
-
;
α
x
,
x
-
η
T
x
,
x
-
∇
f
x
-
P
+
ρ
d
2
x
,
x
-
<
0
,
F
x
,
x
-
;
β
x
,
x
-
η
T
x
,
x
-
∇
g
x
-
P
+
τ
d
2
x
,
x
-
<
0
.Hence (
f
,
g
) is pseudoinvex (
F
,
P
)-type I at u
∈
X, where u
<
x
,
x
∈
X
.Definition 7.
(
f
,
g
) is said to be pseudoinvex F
,
P-type II at x
-
∈
X, if there exists functions b
,
α
i
,
β
j
,
η
,
d and ρ
i
,
τ
j
∈
R
,
P
∈
R
n, such that each x
∈
X; the following inequalities hold:(12)
b
x
,
x
-
f
x
-
f
x
-
≤
0
⟹
F
x
,
x
-
;
α
i
x
,
x
-
η
T
x
,
x
-
∇
f
i
x
-
P
+
ρ
i
d
2
x
,
x
-
<
0
,
i
∈
K
,
F
x
,
x
-
;
β
j
x
,
x
-
η
T
x
,
x
-
∇
g
j
x
-
P
+
τ
j
d
2
x
,
x
-
<
0
,
j
∈
M
.Definition 8.
(
f
,
g
) is said to be pseudoquasi-invex (
F
,
P
)-type I at x
-
∈
X, if there exists functions b
,
α
i
,
β
j
,
η
,
d and ρ
i
,
τ
j
∈
R
,
P
∈
R
n, such that each x
∈
X; the following inequalities hold:(13)
b
x
,
x
-
f
x
-
f
x
-
<
0
⟹
F
x
,
x
-
;
α
i
x
,
x
-
η
T
x
,
x
-
∇
f
i
x
-
P
+
ρ
i
d
2
x
,
x
-
<
0
,
i
∈
K
,
F
x
,
x
-
;
β
j
x
,
x
-
η
T
x
,
x
-
∇
g
j
x
-
P
+
τ
j
d
2
x
,
x
-
≦
0
,
j
∈
M
.Example 9.
Letf
:
R
2
→
R
2, g
:
R
2
→
R, and b
:
R
2
×
R
2
→
R
+
∖
{
0
} be defined by(14)
f
1
x
=
x
1
e
2
x
2
,
f
2
x
=
x
2
3
x
1
2
+
1
,
g
x
=
2
x
1
+
x
2
2
-
3
,
b
x
,
x
-
=
x
1
2
+
x
1
+
1
,and the functions F
:
R
2
×
R
2
×
R
2
→
R and  η
:
R
2
×
R
2
→
R
2 be given by (15)
F
x
,
x
-
;
a
=
a
T
c
x
,
x
-
,
where
c
x
,
x
-
=
x
1
-
x
2
,
x
2
T
is a vector function
.
η
x
,
x
-
=
x
1
2
+
x
2
2
,
x
1
-
2
x
2
+
1
T
.Moreover, the functions α
i
:
R
2
×
R
2
→
R
+
∖
{
0
},  β
:
R
2
×
R
2
→
R
+
∖
{
0
}, and d
:
R
2
×
R
2
→
R are defined by (16)
α
1
x
,
x
-
=
α
2
x
,
x
-
=
x
1
2
+
e
cos
⁡
x
2
,
β
x
,
x
-
=
1
,
d
x
,
x
-
=
x
1
+
x
2
.Then, let (17)
ρ
1
=
ρ
2
=
τ
=
-
1
,
P
=
1,1
T
.It is easy to see that (
f
,
g
) is pseudoquasi-invex F
,
P-type I at the point x
-
=
(
0,0
) with respect to b
,
α
i
(
i
=
1,2
)
,
β
,
η
,
ρ
i
i
=
1,2
,
τ
,
P, and d.Definition 10.
(
f
,
g
) is said to be pseudoquasi-invex (
F
,
P
)-type II at x
-
∈
X, if there exists functions b
,
α
i
,
β
j
,
η
,
d, and ρ
i
,
τ
j
∈
R
,
P
∈
R
n, such that each x
∈
X; the following inequalities hold:(18)
b
x
,
x
-
f
x
-
f
x
-
≤
0
⟹
F
x
,
x
-
;
α
i
x
,
x
-
η
T
x
,
x
-
∇
f
i
x
-
P
+
ρ
i
d
2
x
,
x
-
<
0
,
i
∈
K
,
F
x
,
x
-
;
β
j
x
,
x
-
η
T
x
,
x
-
∇
g
j
x
-
P
+
τ
j
d
2
x
,
x
-
≦
0
,
j
∈
M
.Definition 11.
(
f
,
g
) is said to be quasipseudo-invex (
F
,
P
)-type I at x
-
∈
X, if there exists functions b
,
α
i
,
β
j
,
η
,
d, and ρ
i
,
τ
j
∈
R
,
P
∈
R
n, such that each x
∈
X; the following inequalities hold:(19)
b
x
,
x
-
f
x
-
f
x
-
<
0
⟹
F
x
,
x
-
;
α
i
x
,
x
-
η
T
x
,
x
-
∇
f
i
x
-
P
+
ρ
i
d
2
x
,
x
-
≦
0
,
i
∈
K
,
F
x
,
x
-
;
β
j
x
,
x
-
η
T
x
,
x
-
∇
g
j
x
-
P
+
τ
j
d
2
x
,
x
-
<
0
,
j
∈
M
.Definition 12.
(
f
,
g
) is said to be quasipseudo-invex (
F
,
P
)-type II at x
-
∈
X, if there exists functions b
,
α
i
,
β
j
,
η
,
d, and ρ
i
,
τ
j
∈
R
,
P
∈
R
n, such that each x
∈
X; the following inequalities hold:(20)
b
x
,
x
-
f
x
-
f
x
-
≤
0
⟹
F
x
,
x
-
;
α
i
x
,
x
-
η
T
x
,
x
-
∇
f
i
x
-
P
+
ρ
i
d
2
x
,
x
-
≦
0
,
i
∈
K
,
F
x
,
x
-
;
β
j
x
,
x
-
η
T
x
,
x
-
∇
g
j
x
-
P
+
τ
j
d
2
x
,
x
-
<
0
,
j
∈
M
.


## 3. Sufficient Optimality Conditions

Now, we establish sufficient optimality conditions for the considered optimization problemM
P under the new invexity.Theorem 13.
Letx
-
∈
R
n be a feasible solution in problem M
P. Suppose that(i)
there existsλ
i
≧
0, ∑
i
=
1
k
λ
i
=
1, μ
j
≧
0, i
∈
K
,
j
∈
M, such that (21)
∑
i
=
1
k
λ
i
∇
f
i
x
-
+
∑
j
=
1
m
μ
j
∇
g
j
x
-
=
0
;
(ii)
(
f
,
g
) is pseudoinvex (
F
,
P
)-type I at x
-;
(iii)
∑
i
=
1
k
λ
i
ρ
i
/
α
i
(
x
,
x
-
)
+
∑
j
=
1
m
μ
j
τ
j
/
β
j
(
x
,
x
-
)
≧
0
.
Thenx
- is a weakly efficient solution for M
P.Proof.
Suppose contrary to the result thatx
- is not a weakly efficient solution to M
P. Then there exists x
∈
X
0 such that (22)
f
x
<
f
x
-
.
Withb
(
x
,
x
-
)
>
0, the above inequality yields(23)
b
x
,
x
-
f
x
-
f
x
-
<
0
,from hypothesis (ii), which implies(24)
F
x
,
x
-
;
α
i
x
,
x
-
η
T
x
,
x
-
∇
f
i
x
-
P
+
ρ
i
d
2
x
,
x
-
<
0
,
i
∈
K
,
F
x
,
x
-
;
β
j
x
,
x
-
η
T
x
,
x
-
∇
g
j
x
-
P
+
τ
j
d
2
x
,
x
-
<
0
,
j
∈
M
.
By sublinearity ofF with α
i
(
x
,
x
-
)
>
0 and β
j
(
x
,
x
-
)
>
0, inequalities (24) yield(25)
F
x
,
x
-
;
η
T
x
,
x
-
∇
f
i
x
-
P
<
-
ρ
i
α
i
x
,
x
-
d
2
x
,
x
-
,
i
∈
K
,
F
x
,
x
-
;
η
T
x
,
x
-
∇
g
j
x
-
P
<
-
τ
j
β
j
x
,
x
-
d
2
x
,
x
-
,
j
∈
M
.
Usingλ
i
≧
0, ∑
i
=
1
k
λ
i
=
1, and μ
j
≧
0,j
∈
M, along with the sublinearity of F, from inequality (25), we get(26)
F
x
,
x
-
;
η
T
x
,
x
-
∑
i
=
1
k
λ
i
∇
f
i
x
-
P
<
-
∑
i
=
1
k
λ
i
ρ
i
α
i
x
,
x
-
d
2
x
,
x
-
,
F
x
,
x
-
;
η
T
x
,
x
-
∑
j
=
1
m
μ
j
∇
g
j
x
-
P
≦
-
∑
j
=
1
m
μ
j
τ
j
β
j
x
,
x
-
d
2
x
,
x
-
.
By the sublinearity ofF, we sum (26) to obtain(27)
F
x
,
x
-
;
η
T
x
,
x
-
∑
i
=
1
k
λ
i
∇
f
i
x
-
+
∑
j
=
1
m
μ
j
∇
g
j
x
-
P
<
-
∑
i
=
1
k
λ
i
ρ
i
α
i
x
,
x
-
+
∑
j
=
1
m
μ
j
τ
j
β
j
x
,
x
-
d
2
x
,
x
-
,from hypothesis (iii), which follows(28)
F
x
,
x
-
;
η
T
x
,
x
-
∑
i
=
1
k
λ
i
∇
f
i
x
-
+
∑
j
=
1
m
μ
j
∇
g
j
x
-
P
<
0
.
On the other hand, the hypothesis (i) implies(29)
F
x
,
x
-
;
η
T
x
,
x
-
∑
i
=
1
k
λ
i
∇
f
i
x
-
+
∑
j
=
1
m
μ
j
∇
g
j
x
-
P
=
0
,which contradicts (28). Hence the conclusion of the theorem is established.Theorem 14.
Letx
-
∈
R
n be a feasible solution in problem M
P. Suppose that(i)
there existsλ
i
≧
0, ∑
i
=
1
k
λ
i
=
1, μ
j
≧
0, i
∈
K, j
∈
M, such that (30)
∑
i
=
1
k
λ
i
∇
f
i
x
-
+
∑
j
=
1
m
μ
j
∇
g
j
x
-
=
0
;
(ii)
(
f
,
g
) is pseudoinvex (
F
,
P
)-type II at x
-;
(iii)
∑
i
=
1
k
λ
i
ρ
i
/
α
i
(
x
,
x
-
)
+
∑
j
=
1
m
μ
j
τ
j
/
β
j
(
x
,
x
-
)
≧
0
.
Thenx
- is an efficient solution for M
P.Proof.
By the way of contradiction, suppose thatx
- is not an efficient solution for M
P. Then there exists x
∈
X
0 such that (31)
f
x
≤
f
x
-
.
Withb
(
x
,
x
-
)
>
0, the above inequality yields(32)
b
x
,
x
-
f
x
-
f
x
-
≤
0
,by hypothesis (ii), which follows(33)
F
x
,
x
-
;
α
i
x
,
x
-
η
T
x
,
x
-
∇
f
i
x
-
P
+
ρ
i
d
2
x
,
x
-
<
0
,
i
∈
K
,
F
x
,
x
-
;
β
j
x
,
x
-
η
T
x
,
x
-
∇
g
j
x
-
P
+
τ
j
d
2
x
,
x
-
<
0
,
j
∈
M
.
Usingα
i
(
x
,
x
-
)
>
0, β
j
(
x
,
x
-
)
>
0 and λ
i
≧
0,  ∑
i
=
1
k
λ
i
=
1, μ
j
≧
0
,
j
∈
M, along with the sublinearity of F, inequality (33) yields(34)
F
x
,
x
-
;
η
T
x
,
x
-
∑
i
=
1
k
λ
i
∇
f
i
x
-
P
<
-
∑
i
=
1
k
λ
i
ρ
i
α
i
x
,
x
-
d
2
x
,
x
-
,
F
x
,
x
-
;
η
T
x
,
x
-
∑
j
=
1
m
μ
j
∇
g
j
x
-
P
≦
-
∑
j
=
1
m
μ
j
τ
j
β
j
x
,
x
-
d
2
x
,
x
-
.
Summing inequalities (34) with the sublinearity of F, we obtain(35)
F
x
,
x
-
;
η
T
x
,
x
-
∑
i
=
1
k
λ
i
∇
f
i
x
-
+
∑
j
=
1
m
μ
j
∇
g
j
x
-
P
<
-
∑
i
=
1
k
λ
i
ρ
i
α
i
x
,
x
-
+
∑
j
=
1
m
μ
j
τ
j
β
j
x
,
x
-
d
2
x
,
x
-
.From assumption (i), we have(36)
∑
i
=
1
k
λ
i
ρ
i
α
i
x
,
x
-
+
∑
j
=
1
m
μ
j
τ
j
β
j
x
,
x
-
d
2
x
,
x
-
<
0
;that is,(37)
∑
i
=
1
k
λ
i
ρ
i
α
i
x
,
x
-
+
∑
j
=
1
m
μ
j
τ
j
β
j
x
,
x
-
<
0
,which contradicts hypothesis (iii). That completes the proof.Theorem 15.
Letx
-
∈
R
n be a feasible solution in problem M
P. Suppose that(i)
there existsλ
i
≧
0, ∑
i
=
1
k
λ
i
=
1, μ
j
≧
0, i
∈
K, j
∈
M, such that (38)
∑
i
=
1
k
λ
i
∇
f
i
x
-
+
∑
j
=
1
m
μ
j
∇
g
j
x
-
=
0
;
(ii)
(
f
,
g
) is pseudoquasi-invex (
F
,
P
)-type I at x
-;
(iii)
∑
i
=
1
k
λ
i
ρ
i
/
α
i
(
x
,
x
-
)
+
∑
j
=
1
m
μ
j
τ
j
/
β
j
(
x
,
x
-
)
≧
0
.
Thenx
- is a weakly efficient solution for M
P.Proof.
The proof follows the lines of Theorem13.Theorem 16.
Letx
-
∈
R
n be a feasible solution in problem M
P. Suppose that(i)
there existsλ
i
≧
0, ∑
i
=
1
k
λ
i
=
1, μ
j
≧
0, i
∈
K, j
∈
M, such that(39)
∑
i
=
1
k
λ
i
∇
f
i
x
-
+
∑
j
=
1
m
μ
j
∇
g
j
x
-
=
0
;
(ii)
(
f
,
g
) is pseudoquasi-invex (
F
,
P
)-type II at x
-;
(iii)
∑
i
=
1
k
λ
i
ρ
i
/
α
i
(
x
,
x
-
)
+
∑
j
=
1
m
μ
j
τ
j
/
β
j
(
x
,
x
-
)
≧
0
.
Thenx
- is an efficient solution for M
P.Proof.
The proof follows the lines of Theorem14.Theorem 17.
Letx
-
∈
R
n be a feasible solution in problem M
P. Suppose that(i)
there existsλ
i
≧
0, ∑
i
=
1
k
λ
i
=
1
,
μ
j
≧
0 (at least one μ
j
>
0), i
∈
K, j
∈
M, such that (40)
∑
i
=
1
k
λ
i
∇
f
i
x
-
+
∑
j
=
1
m
μ
j
∇
g
j
x
-
=
0
;
(ii)
(
f
,
g
) is quasipseudo-invex (
F
,
P
)-type I at x
-;
(iii)
∑
i
=
1
k
λ
i
ρ
i
/
α
i
(
x
,
x
-
)
+
∑
j
=
1
m
μ
j
τ
j
/
β
j
(
x
,
x
-
)
≧
0
.
Thenx
- is a weakly efficient solution for M
P.Proof.
Suppose contrary to the result thatx
- is not a weakly efficient solution to M
P. Then there exists x
∈
X
0 such that (41)
f
x
<
f
x
-
,with b
(
x
,
x
-
)
>
0; the above inequality yields(42)
b
x
,
x
-
f
x
-
f
x
-
<
0
,from hypothesis (ii), which implies(43)
F
x
,
x
-
;
α
i
x
,
x
-
η
T
x
,
x
-
∇
f
i
x
-
P
+
ρ
i
d
2
x
,
x
-
≦
0
,
i
∈
K
,
F
x
,
x
-
;
β
j
x
,
x
-
η
T
x
,
x
-
∇
g
j
x
-
P
+
τ
j
d
2
x
,
x
-
<
0
,
j
∈
M
.
By sublinearity ofF with α
i
(
x
,
x
-
)
>
0 and β
j
(
x
,
x
-
)
>
0, inequalities (43) yield(44)
F
x
,
x
-
;
η
T
x
,
x
-
∇
f
i
x
-
P
≦
-
ρ
i
α
i
x
,
x
-
d
2
x
,
x
-
,
i
∈
K
,
(45)
F
x
,
x
-
;
η
T
x
,
x
-
∇
g
j
x
-
P
<
-
τ
j
β
j
x
,
x
-
d
2
x
,
x
-
,
j
∈
M
.
Usingλ
i
≧
0,  ∑
i
=
1
k
λ
i
=
1 along with the sublinearity of F, from inequality (44), we get(46)
F
x
,
x
-
;
η
T
x
,
x
-
∑
i
=
1
k
λ
i
∇
f
i
x
-
P
≦
-
∑
i
=
1
k
λ
i
ρ
i
α
i
x
,
x
-
d
2
x
,
x
-
.
Withμ
j
≧
0 (at least one μ
j
>
0),  j
∈
M, and using the sublinearity of F, inequality (45) follows(47)
F
x
,
x
-
;
η
T
x
,
x
-
∑
j
=
1
m
μ
j
∇
g
j
x
-
P
<
-
∑
j
=
1
m
μ
j
τ
j
β
j
x
,
x
-
d
2
x
,
x
-
.
By the sublinearity ofF, we sum (46) and (47) to obtain(48)
F
x
,
x
-
;
η
T
x
,
x
-
∑
i
=
1
k
λ
i
∇
f
i
x
-
+
∑
j
=
1
m
μ
j
∇
g
j
x
-
P
<
-
∑
i
=
1
k
λ
i
ρ
i
α
i
x
,
x
-
+
∑
j
=
1
m
μ
j
τ
j
β
j
x
,
x
-
d
2
x
,
x
-
,from hypothesis (iii), which follows(49)
F
x
,
x
-
;
η
T
x
,
x
-
∑
i
=
1
k
λ
i
∇
f
i
x
-
+
∑
j
=
1
m
μ
j
∇
g
j
x
-
P
<
0
.
On the other hand, the hypothesis (i) implies(50)
F
x
,
x
-
;
η
T
x
,
x
-
∑
i
=
1
k
λ
i
∇
f
i
x
-
+
∑
j
=
1
m
μ
j
∇
g
j
x
-
P
=
0
,which contradicts (49). Hence the conclusion of theorem is established.Example 18.
We consider the following programming problem:P
Minimize
f
x
=
x
3
subject to
g
x
=
1
-
e
x
≦
0
,
x
∈
R
.
The set of all feasible solutions ofP can be given by X
0
=
{
x
∣
x
≧
0
}.
Again, letF
(
x
,
x
-
;
a
) be the function defined by F
(
x
,
x
-
;
a
)
=
a
(
2
x
-
+
1
)
.
It can be verified that(
f
,
g
) is quasipseudo-invex F
,
P-type I at x
-
=
0 with α
(
x
,
x
-
)
=
1
/
x
2
+
1,  β
x
,
x
-
=
2
/
x
2
+
1,  η
x
,
x
-
=
x
2
+
1,  P
=
1,  d
x
,
x
-
=
x
-
-
1, and ρ
=
0, τ
=
2
/
3,  ∇
f
(
0
)
=
0,  ∇
g
(
0
)
=
-
1
.
Clearly,x
-
=
0 is a feasible solution for problem P and it satisfied the assumptions of Theorem 17, as there exist λ
=
1,  μ
=
0, such that(51)
λ
∇
f
0
+
μ
∇
g
0
=
0
,
λ
ρ
α
x
,
x
-
+
μ
τ
β
x
,
x
-
=
0
.
We observe that there exists no otherx
∈
X
0, such that f
(
x
)
<
f
(
x
-
). Hence, x
-
=
0 is an efficient solution for P.Similarly, we can establish the following theorem.Theorem 19.
Letx
-
∈
R
n be a feasible solution in problem M
P. Suppose that(i)
there existsλ
i
≧
0, ∑
i
=
1
k
λ
i
=
1
,
μ
j
≧
0  (at least one μ
j
>
0, as j
∈
J
-
(
x
-
)), i
∈
K, j
∈
M, such that (52)
∑
i
=
1
k
λ
i
∇
f
i
x
-
+
∑
j
=
1
m
μ
j
∇
g
j
x
-
=
0
;
(ii)
(
f
,
g
) is quasipseudo-invex F
,
P-type II at x
-;
(iii)
∑
i
=
1
k
λ
i
ρ
i
/
α
i
(
x
,
x
-
)
+
∑
j
=
1
m
μ
j
τ
j
/
β
j
(
x
,
x
-
)
≧
0
.
Thenx
- is an efficient solution for M
P.


## 4. Mond-Weir Duality

In this section, a dual problem is considered for the class of multiobjective programming problem with the new invex functions.Consider the following Mond-Weir dual problem related to problemM
P:M
D
I
Maximize
f
u
=
f
1
u
,
f
2
u
,
…
,
f
k
u
subject to
∑
i
=
1
k
λ
-
i
∇
f
i
u
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
=
0
,
λ
-
i
≧
0
,
∑
i
=
1
k
λ
-
i
=
1
,
i
∈
K
,
μ
-
j
≧
0
,
j
∈
M
.LetW
1
=
{
u
,
λ
-
,
μ
-
∈
X
×
R
k
×
R
m
:
∑
i
=
1
k
λ
-
i
∇
f
i
(
u
)
+
∑
j
=
1
m
μ
-
j
∇
g
j
(
u
)
=
0,  λ
-
≥
0,  ∑
i
=
1
k
λ
-
i
=
1
,
μ
-
≧
0
} be the set of all feasible solutions in problem M
D
I.Theorem 20 (weak duality).
Letx and (
u
,
λ
-
,
μ
-
) be feasible solutions for M
P and M
D
I, respectively. Moreover, assume that (53)
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
,
u
+
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
,
u
≧
0
.
If one of the following conditions is satisfied:(a)
f
,
g is pseudoinvex (
F
,
P
)-type I at u,
(b)
f
,
g is pseudoquasi-invex (
F
,
P
)-type I at u,
then the following can not hold:(54)
f
x
<
f
u
.Proof.
Suppose contrary to the result that(55)
f
x
<
f
uhold.
Byb
(
x
,
u
)
>
0, the above inequality follows(56)
b
x
,
u
f
x
-
f
u
<
0
;by assumption (a), (
f
,
g
) is pseudoinvex (
F
,
P
)-type I at u, which yields(57)
F
x
,
u
;
α
i
x
,
u
η
T
x
,
u
∇
f
i
u
P
+
ρ
i
d
2
x
,
u
<
0
,
i
∈
K
,
F
x
,
u
;
β
j
x
,
u
η
T
x
,
u
∇
g
j
u
P
+
τ
j
d
2
x
,
u
<
0
,
j
∈
M
.
By sublinearity ofF together with α
i
(
x
,
u
)
>
0 and β
j
(
x
,
u
)
>
0, the above inequalities yield(58)
F
x
,
u
;
η
T
x
,
u
∇
f
i
u
P
<
-
ρ
i
α
i
x
,
u
d
2
x
,
u
,
i
∈
K
,
F
x
,
u
;
η
T
x
,
u
∇
g
j
u
P
<
-
τ
j
β
j
x
,
u
d
2
x
,
u
,
j
∈
M
.
From the feasibility of(
u
,
λ
-
,
μ
-
) in Mond-Weir dual problem M
D
I, it follows that λ
-
i
≧
0,  ∑
i
=
1
k
λ
-
i
=
1, μ
-
j
≧
0, j
∈
M. Multiplying inequalities (58) by λ
-
i and μ
-
j, respectively, together with the sublinearity of F, we get(59)
F
x
,
u
;
η
T
x
,
u
∑
i
=
1
k
λ
-
i
∇
f
i
u
P
<
-
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
,
u
d
2
x
,
u
,
F
x
,
u
;
η
T
x
,
u
∑
j
=
1
m
μ
-
j
∇
g
j
u
P
≦
-
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
,
u
d
2
x
,
u
.Adding both sides of (59) with the sublinearity of F, we obtain(60)
F
x
,
u
;
η
T
x
,
u
∑
i
=
1
k
λ
-
i
∇
f
i
u
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
P
<
-
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
,
u
+
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
,
u
d
2
x
,
u
.
From assumption,∑
i
=
1
k
λ
-
i
ρ
i
/
α
i
(
x
,
u
)
+
∑
j
=
1
m
μ
-
j
τ
j
/
β
j
(
x
,
u
)
≧
0; inequality (60) implies(61)
F
x
,
u
;
η
T
x
,
u
∑
i
=
1
k
λ
-
i
∇
f
i
u
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
P
<
0
.
By the constraint condition of dual problemM
D
I and F
(
x
,
u
;
0
)
=
0, we have(62)
F
x
,
u
;
η
T
x
,
u
∑
i
=
1
k
λ
-
i
∇
f
i
u
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
P
=
0
,which contradicts inequality (61). Thus, the conclusion of theorem holds.
The proof of part (b) is similar to the proof of part (a).Theorem 21 (weak duality).
Letx and (
u
,
λ
-
,
μ
-
) be feasible solutions for M
P and M
D
I, respectively. Moreover, assume that (63)
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
,
u
+
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
,
u
≧
0
.
If one of the following conditions is satisfied:(a)
(
f
,
g
) is pseudoinvex (
F
,
P
)-type II at u,
(b)
(
f
,
g
) is pseudoquasi-invex (
F
,
P
)-type II at u,
then the following can not hold:(64)
f
x
≤
f
u
.Proof.
Suppose contrary to the result that(65)
f
x
≤
f
uhold.
Byb
(
x
,
u
)
>
0, the above inequality yields(66)
b
x
,
u
f
x
-
f
u
≤
0
;by assumption (a), (
f
,
g
) is pseudoinvex (
F
,
P
)-type II at u, which yields(67)
F
x
,
u
;
α
i
x
,
u
η
T
x
,
u
∇
f
i
u
P
+
ρ
i
d
2
x
,
u
<
0
,
i
∈
K
,
F
x
,
u
;
β
j
x
,
u
η
T
x
,
u
∇
g
j
u
P
+
τ
j
d
2
x
,
u
<
0
,
j
∈
M
.
By sublinearity ofF together with α
i
x
,
u
>
0 and β
j
(
x
,
u
)
>
0, the above inequalities yield(68)
F
x
,
u
;
η
T
x
,
u
∇
f
i
u
P
<
-
ρ
i
α
i
x
,
u
d
2
x
,
u
,
i
∈
K
,
F
x
,
u
;
η
T
x
,
u
∇
g
j
u
P
<
-
τ
j
β
j
x
,
u
d
2
x
,
u
,
j
∈
M
.
From the feasibility ofu
,
λ
-
,
μ
- in Mond-Weir dual problem M
D
I, we have(69)
∑
i
=
1
k
λ
-
i
∇
f
i
u
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
=
0
.
Forλ
-
i
≧
0, ∑
i
=
1
k
λ
-
i
=
1,  μ
-
j
≧
0,  j
∈
M. Multiplying inequalities (68) by λ
-
i and μ
-
j, respectively, together with the sublinearity of F, we have(70)
F
x
,
u
;
η
T
x
,
u
∑
i
=
1
k
λ
-
i
∇
f
i
u
P
<
-
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
,
u
d
2
x
,
u
,
F
x
,
u
;
η
T
x
,
u
∑
j
=
1
m
μ
-
j
∇
g
j
u
P
≦
-
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
,
u
d
2
x
,
u
.
Adding both sides of (70) with the sublinearity of F, we get(71)
F
x
,
u
;
η
T
x
,
u
∑
i
=
1
k
λ
-
i
∇
f
i
u
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
P
<
-
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
,
u
+
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
,
u
d
2
x
,
u
.
Combining (69) and (71), we obtain(72)
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
,
u
+
∑
j
∈
J
-
u
μ
-
j
τ
j
β
j
x
,
u
<
0
,which contradicts the assumption(73)
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
,
u
+
∑
j
∈
J
-
u
μ
-
j
τ
j
β
j
x
,
u
≧
0
.
Thus, the conclusion of the theorem holds.
The proof of part (b) is similar to the proof of part (a).Theorem 22 (strong duality).
Assume thatx
- is a weakly efficient solution of M
P. Suppose that there exists λ
-
≥
0 and μ
-
≧
0, such that (
x
-
,
λ
-
,
μ
-
) is feasible for M
D
I. Furthermore, if the weak duality Theorem 20 holds for all feasible solutions of the problems M
P and M
D
I, then (
x
-
,
λ
-
,
μ
-
) is a weakly efficient solution of M
D
I.Proof.
Suppose that(
x
-
,
λ
-
,
μ
-
) is not a weakly efficient solution of M
D
I; then there exists another feasible solution (
u
,
λ
,
μ
) of M
D
I such that (74)
f
x
-
<
f
u
,which is a contradiction to Theorem 20. Hence (
x
-
,
λ
-
,
μ
-
) is a weakly efficient solution of M
D
I.Theorem 23 (strong duality).
Assume thatx
- is an efficient solution of M
P. Suppose that there exist λ
-
≥
0 and μ
-
≧
0, such that (
x
-
,
λ
-
,
μ
-
) is feasible for M
D
I. Furthermore, if the weak duality Theorem 21 holds for all feasible solutions of the problems M
P and M
D
I, then (
x
-
,
λ
-
,
μ
-
) is an efficient solution of M
D
I.Proof.
Suppose thatx
-
,
λ
-
,
μ
- is not an efficient solution of M
D
I; then there exists another feasible solution u
,
λ
,
μ of M
D
I such that (75)
f
x
-
≤
f
u
,which is a contradiction to Theorem 21. Hence x
-
,
λ
-
,
μ
- is an efficient solution of M
D
I.Theorem 24 (strict converse duality).
Letx
- and u
-
,
λ
-
,
μ
- be feasible solutions for M
P and M
D
I, respectively. Suppose that f
x
-
≤
f
u
-, and ∑
i
=
1
k
λ
-
i
ρ
i
/
α
i
(
x
-
,
u
-
)
+
∑
j
=
1
m
μ
-
j
τ
j
/
β
j
(
x
-
,
u
-
)
≧
0. If one of the following conditions is satisfied:(a)
f
,
g is pseudoinvex F
,
P-type II at u
-,
(b)
f
,
g is pseudoquasi-invex F
,
P-type II at u
-,
thenx
-
=
u
-.Proof.
Suppose thatx
-
≠
u
-.
Byb
x
-
,
u
-
>
0, the condition f
(
x
-
)
≤
f
(
u
-
) yields(76)
b
x
-
,
u
-
f
x
-
-
f
u
-
≤
0
,using assumption (a), f
,
g is pseudoinvex F
,
P-type II at u
-, which yields(77)
F
x
-
,
u
-
;
α
i
x
-
,
u
-
η
T
x
-
,
u
-
∇
f
i
u
-
P
+
ρ
i
d
2
x
-
,
u
-
<
0
,
i
∈
K
,
F
x
-
,
u
-
;
β
j
x
-
,
u
-
η
T
x
-
,
u
-
∇
g
j
u
-
P
+
τ
j
d
2
x
-
,
u
-
<
0
,
j
∈
M
.
By sublinearity ofF together with α
i
x
-
,
u
-
>
0 and β
j
x
-
,
u
-
>
0, the above inequalities yield(78)
F
x
-
,
u
-
;
η
T
x
-
,
u
-
∇
f
i
u
-
P
<
-
ρ
i
α
i
x
-
,
u
-
d
2
x
-
,
u
-
,
i
∈
K
,
F
x
-
,
u
-
;
η
T
x
-
,
u
-
∇
g
j
u
-
P
<
-
τ
j
β
j
x
-
,
u
-
d
2
x
-
,
u
-
,
j
∈
M
.
Becauseu
-
,
λ
-
,
μ
- is feasible for M
D
I, then(79)
∑
i
=
1
k
λ
-
i
∇
f
i
u
-
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
-
=
0
.
Forλ
-
i
≧
0
,
∑
i
=
1
k
λ
-
i
=
1
,
μ
-
j
≧
0
,
j
∈
M. Multiplying inequalities (78) by λ
-
i and μ
-
j, respectively, together with the sublinearity of F, we have(80)
F
x
-
,
u
-
;
η
T
x
-
,
u
-
∑
i
=
1
k
λ
-
i
∇
f
i
u
-
P
<
-
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
-
,
u
-
d
2
x
-
,
u
-
,
F
x
-
,
u
-
;
η
T
x
-
,
u
-
∑
j
=
1
m
μ
-
j
∇
g
j
u
-
P
≦
-
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
-
,
u
-
d
2
x
-
,
u
-
.
Adding both sides of (80) with the sublinearity of F, we get(81)
F
x
-
,
u
-
;
η
T
x
-
,
u
-
∑
i
=
1
k
λ
-
i
∇
f
i
u
-
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
-
P
<
-
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
-
,
u
-
+
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
-
,
u
-
d
2
x
-
,
u
-
.
Using the assumption∑
i
=
1
k
λ
-
i
ρ
i
/
α
i
(
x
-
,
u
-
)
+
∑
j
=
1
m
μ
-
j
τ
j
/
β
j
(
x
-
,
u
-
)
≧
0, inequality (81) follows that (82)
F
x
-
,
u
-
;
η
T
x
-
,
u
-
∑
i
=
1
k
λ
-
i
∇
f
i
u
-
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
-
P
<
0
.
Using inequality (79) together with the sublinearity of F, we obtain (83)
F
x
-
,
u
-
;
η
T
x
-
,
u
-
∑
i
=
1
k
λ
-
i
∇
f
i
u
-
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
-
P
=
0
,which is a contradiction to (82). Then x
-
=
u
-.
The proof of part (b) is similar to the proof of part (a).Theorem 25 (strict converse duality).
Letx
- and u
-
,
λ
-
,
μ
- be feasible solutions for M
P and M
D
I, respectively. Suppose that f
x
-
<
f
u
-, and ∑
i
=
1
k
λ
-
i
ρ
i
/
α
i
(
x
-
,
u
-
)
+
∑
j
=
1
m
μ
-
j
τ
j
/
β
j
(
x
-
,
u
-
)
≧
0. If one of the following conditions is satisfied:(a)
f
,
g is pseudoinvex F
,
P-type I at u
-,
(b)
f
,
g is pseudoquasi-invex F
,
P-type I at u
-,
thenx
-
=
u
-.Proof.
Suppose thatx
-
≠
u
-.
Byb
x
-
,
u
-
>
0, the condition f
x
-
<
f
u
- yields(84)
b
x
-
,
u
-
f
x
-
-
f
u
-
<
0
;using assumption (a), f
,
g is pseudoinvex F
,
P-type I at u
-, which yields(85)
F
x
-
,
u
-
;
α
i
x
-
,
u
-
η
T
x
-
,
u
-
∇
f
i
u
-
P
+
ρ
i
d
2
x
-
,
u
-
<
0
,
i
∈
K
,
F
x
-
,
u
-
;
β
j
x
-
,
u
-
η
T
x
-
,
u
-
∇
g
j
u
-
P
+
τ
j
d
2
x
-
,
u
-
<
0
,
j
∈
M
.
By using the sublinearity ofF with α
i
x
-
,
u
-
>
0 and β
j
x
-
,
u
-
>
0 and u
-
,
λ
-
,
μ
-
∈
W
1, inequalities (85) imply(86)
F
x
-
,
u
-
;
η
T
x
-
,
u
-
∑
i
=
1
k
λ
-
i
∇
f
i
u
-
P
<
-
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
-
,
u
-
d
2
x
-
,
u
-
,
F
x
-
,
u
-
;
η
T
x
-
,
u
-
∑
j
=
1
m
μ
-
j
∇
g
j
u
-
P
≦
-
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
-
,
u
-
d
2
x
-
,
u
-
.
Adding both sides of (86) with the sublinearity of F, we have(87)
F
x
-
,
u
-
;
η
T
x
-
,
u
-
∑
i
=
1
k
λ
-
i
∇
f
i
u
-
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
-
P
<
-
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
-
,
u
-
+
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
-
,
u
-
d
2
x
-
,
u
-
.
Using the condition∑
i
=
1
k
λ
-
i
ρ
i
/
α
i
(
x
-
,
u
-
)
+
∑
j
=
1
m
μ
-
j
τ
j
/
β
j
(
x
-
,
u
-
)
≧
0, inequality (87) yields (88)
F
x
-
,
u
-
;
η
T
x
-
,
u
-
∑
i
=
1
k
λ
-
i
∇
f
i
u
-
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
-
P
<
0
.
According to the constraint condition ofM
D
I with F
x
-
,
u
-
;
0
=
0, we get (89)
F
x
-
,
u
-
;
η
T
x
-
,
u
-
∑
i
=
1
k
λ
-
i
∇
f
i
u
-
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
-
P
=
0
,which is a contradiction to (88). Then x
-
=
u
-.
The proof of part (b) is similar to the proof of part (a).


## 5. Wolfe Type Duality

In this section, we consider the Wolfe type dual forM
P and establish various duality theorems. Let e be the vector of R
k whose components are all ones.M
D
I
I
Maximize
G
u
=
f
u
+
μ
-
T
g
u
e
=
f
1
u
+
∑
j
=
1
m
μ
-
j
g
j
u
,
f
2
u
+
∑
j
=
1
m
μ
-
j
g
j
u
,
…
,
f
k
u
+
∑
j
=
1
m
μ
-
j
g
j
u
subject to
∑
i
=
1
k
λ
-
i
∇
f
i
u
+
∑
j
=
1
m
t
-
j
∇
g
j
u
=
0
,
t
-
j
=
2
μ
-
j
,
j
∈
M
,
λ
-
i
≧
0
,
∑
i
=
1
k
λ
-
i
=
1
,
i
∈
K
,
μ
-
j
≧
0
,
j
∈
M
.LetW
2
=
{
u
,
λ
-
,
μ
-
,
t
-
∈
X
×
R
k
×
R
m
×
R
m
:
∑
i
=
1
k
λ
-
i
∇
f
i
u
+
∑
j
=
1
m
t
-
j
g
j
u
=
0,  λ
-
≥
0,  ∑
i
=
1
k
λ
-
i
=
1
,
μ
-
≧
0
,
t
-
≧
0
} be the set of all feasible solutions in problem M
D
I
I.Theorem 26 (weak duality).
Letx
∈
X
0 and u
,
λ
-
,
μ
-
,
t
-
∈
W
2. Suppose that (90)
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
,
u
+
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
,
u
≧
0
.
If one of the following conditions is satisfied:(a)
G
,
g is pseudoinvex F
,
P-type I at u,
(b)
G
,
g is pseudoquasi-invex F
,
P-type I at u,then the following can not hold:(91)
f
x
<
G
u
.Proof.
If(92)
f
x
<
G
uholds, then we have(93)
f
x
<
f
u
+
μ
-
T
g
u
e
.
Sincex
∈
X
0 and μ
-
≧
0, the above inequality yields(94)
f
x
+
μ
-
T
g
x
e
<
f
u
+
μ
-
T
g
u
e
;that is,(95)
G
x
<
G
u
;by using b
x
,
u
>
0, we have(96)
b
x
,
u
G
x
-
G
u
<
0
;by assumption (a), G
,
g is pseudoinvex F
,
P-type I at u, which implies (97)
F
x
,
u
;
α
i
x
,
u
η
T
x
,
u
∇
f
i
u
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
P
+
ρ
i
d
2
x
,
u
<
0
,
i
∈
K
,
F
x
,
u
;
β
j
x
,
u
η
T
x
,
u
∇
g
j
u
P
+
τ
j
d
2
x
,
u
<
0
,
j
∈
M
.
By sublinearity ofF together with α
i
x
,
u
>
0, β
j
(
x
,
u
)
>
0, and the feasibility of u
,
λ
-
,
μ
-
,
t
- in M
D
I
I, the above inequalities yield(98)
F
x
,
u
;
η
T
x
,
u
∑
i
=
1
k
λ
-
i
∇
f
i
u
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
P
<
-
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
,
u
d
2
x
,
u
,
F
x
,
u
;
η
T
x
,
u
∑
j
=
1
m
μ
-
j
∇
g
j
u
P
≦
-
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
,
u
d
2
x
,
u
.
Adding both sides of (98) with the sublinearity of F, we get(99)
F
x
,
u
;
η
T
x
,
u
∑
i
=
1
k
λ
-
i
∇
f
i
u
+
∑
j
=
1
m
t
-
j
∇
g
j
u
P
<
-
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
,
u
+
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
,
u
d
2
x
,
u
,
t
-
j
=
2
μ
-
j
,
j
∈
M
.
From assumption,∑
i
=
1
k
λ
-
i
ρ
i
/
α
i
(
x
,
u
)
+
∑
j
=
1
m
μ
-
j
τ
j
/
β
j
(
x
,
u
)
≧
0; inequality (99) implies(100)
F
x
,
u
;
η
T
x
,
u
∑
i
=
1
k
λ
-
i
∇
f
i
u
+
∑
j
=
1
m
t
-
j
∇
g
j
u
P
<
0
.
By the constraint condition ofM
D
I
I and F
x
,
u
;
0
=
0, we have(101)
F
x
,
u
;
η
T
x
,
u
∑
i
=
1
k
λ
-
i
∇
f
i
u
+
∑
j
=
1
m
t
-
j
∇
g
j
u
P
=
0
,which contradicts inequality (100). Thus, the conclusion of theorem holds.
The proof of part (b) is similar to the proof of part (a).Theorem 27 (weak duality).
Letx
∈
X
0 and u
,
λ
-
,
μ
-
,
t
-
∈
W
2. Suppose that (102)
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
,
u
+
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
,
u
≧
0
.
If one of the following conditions is satisfied:(a)
G
,
g is pseudoinvex F
,
P-type II at u,
(b)
G
,
g is pseudoquasi-invex F
,
P-type II at u,
then the following can not hold:(103)
f
x
≤
G
u
.Proof.
Suppose contrary to the result that(104)
f
x
≤
G
uhold. That is,(105)
f
x
≤
f
u
+
μ
-
T
g
u
e
.
Sincex
∈
X
0 and μ
-
≧
0, the above inequality yields(106)
f
x
+
μ
-
T
g
x
e
≤
f
u
+
μ
-
T
g
u
e
;that is,(107)
G
x
≤
G
u
.
By usingb
(
x
,
u
)
>
0, we get(108)
b
x
,
u
G
x
-
G
u
≤
0
.
By assumption (a),G
,
g is pseudoinvex F
,
P-type II at u, which yields (109)
F
x
,
u
;
α
i
x
,
u
η
T
x
,
u
∇
f
i
u
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
P
+
ρ
i
d
2
x
,
u
<
0
,
i
∈
K
,
F
x
,
u
;
β
j
x
,
u
η
T
x
,
u
∇
g
j
u
P
+
τ
j
d
2
x
,
u
<
0
,
j
∈
M
.
By sublinearity ofF together with α
i
(
x
,
u
)
>
0,β
j
(
x
,
u
)
>
0, and the feasibility of u
,
λ
-
,
μ
-
,
t
- in M
D
I
I, the above inequalities yield(110)
F
x
,
u
;
η
T
x
,
u
∑
i
=
1
k
λ
-
i
∇
f
i
u
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
P
<
-
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
,
u
d
2
x
,
u
,
F
x
,
u
;
η
T
x
,
u
∑
j
=
1
m
μ
-
j
∇
g
j
u
P
≦
-
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
,
u
d
2
x
,
u
.
Adding both sides of (110) with the sublinearity of F, we get(111)
F
x
,
u
;
η
T
x
,
u
∑
i
=
1
k
λ
-
i
∇
f
i
u
+
∑
j
=
1
m
t
-
j
∇
g
j
u
P
<
-
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
,
u
+
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
,
u
d
2
x
,
u
,
t
-
j
=
2
μ
-
j
,
j
∈
M
.
By using the constraint condition ofM
D
I
I and F
x
,
u
;
0
=
0, inequality (111) implies(112)
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
,
u
+
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
,
u
<
0
,which contradicts the assumption ∑
i
=
1
k
λ
-
i
ρ
i
/
α
i
(
x
,
u
)
+
∑
j
=
1
m
μ
-
j
τ
j
/
β
j
(
x
,
u
)
≧
0.
Thus, the conclusion of theorem holds.
The proof of part (b) is similar to the proof of part (a).Theorem 28 (strong duality).
Assume thatx
- is a weakly efficient solution of M
P. Suppose that there exist λ
-
≥
0, μ
-
≧
0, and t
-
≧
0, such that x
-
,
λ
-
,
μ
-
,
t
- is feasible for M
D
I
I. Furthermore, if the weak duality Theorem 26 holds for all feasible solutions of the problems M
P and M
D
I
I, then x
-
,
λ
-
,
μ
-
,
t
- is a weakly efficient solution of M
D
I
I.Proof.
Suppose thatx
-
,
λ
-
,
μ
-
,
t
- is not a weakly efficient solution of M
D
I
I; then there exists another feasible solution x
,
λ
,
μ
,
t of M
D
I
I such that (113)
G
x
-
<
G
u
;that is,(114)
f
x
-
+
μ
-
T
g
x
-
e
<
f
u
+
μ
T
g
u
e
.
Forx
-
∈
X
0 and x
-
,
λ
-
,
μ
-
,
t
-
∈
W
2, the above inequality yields(115)
f
x
-
<
f
u
+
μ
T
g
u
e
;that is,(116)
f
x
-
<
G
u
,which is a contradiction to Theorem 26. Hence x
-
,
λ
-
,
μ
-
,
t
- is a weakly efficient solution of M
D
I
I.Theorem 29 (strict converse duality).
Letx
- and u
-
,
λ
-
,
μ
-
,
t
- be feasible solutions for M
P and M
D
I
I, respectively. Suppose that f
(
x
-
)
<
G
u
-, and ∑
i
=
1
k
λ
-
i
ρ
i
/
α
i
(
x
-
,
u
-
)
+
∑
j
=
1
m
μ
-
j
τ
j
/
β
j
(
x
-
,
u
-
)
≧
0. If one of the following conditions is satisfied:(a)
f
,
g is pseudoinvex F
,
P-type I at u
-,
(b)
f
,
g is pseudoquasi-invex F
,
P-type I at u
-,
thenx
-
=
u
-.Proof.
Suppose thatx
-
≠
u
-.
Fromx
-
∈
X
0 and u
-
,
λ
-
,
μ
-
,
t
-
∈
W
2, the condition f
x
-
<
G
u
- yields(117)
f
x
-
+
μ
-
T
g
x
-
e
<
f
u
-
+
μ
-
T
g
u
-
e
;that is,(118)
G
x
-
<
G
u
-
,by b
x
-
,
u
-
>
0, which implies(119)
b
x
-
,
u
-
G
x
-
-
G
u
-
<
0
;using assumption (a), G
,
g is pseudoinvex F
,
P-type I at u
-, which yields(120)
F
x
-
,
u
-
;
α
i
x
-
,
u
-
η
T
x
-
,
u
-
∇
f
i
u
-
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
-
P
+
ρ
i
d
2
x
-
,
u
-
<
0
,
i
∈
K
;
F
x
-
,
u
-
;
β
j
x
-
,
u
-
η
T
x
-
,
u
-
∇
g
j
u
-
P
+
τ
j
d
2
x
-
,
u
-
<
0
,
j
∈
M
.
By sublinearity ofF together with α
i
x
-
,
u
-
>
0,β
j
x
-
,
u
-
>
0, and u
-
,
λ
-
,
μ
-
,
t
-
∈
W
2, the above inequalities yield(121)
F
x
-
,
u
-
;
η
T
x
-
,
u
-
∑
i
=
1
k
λ
-
i
∇
f
i
u
-
+
∑
j
=
1
m
μ
-
j
∇
g
j
u
-
P
<
-
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
-
,
u
-
d
2
x
-
,
u
-
,
F
x
-
,
u
-
;
η
T
x
-
,
u
-
∑
j
=
1
m
μ
-
j
∇
g
j
u
-
P
≦
-
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
-
,
u
-
d
2
x
-
,
u
-
.
Adding both sides of (121) with the sublinearity of F, we have(122)
F
x
-
,
u
-
;
η
T
x
-
,
u
-
∑
i
=
1
k
λ
-
i
∇
f
i
u
-
+
∑
j
=
1
m
t
-
j
∇
g
j
u
-
P
<
-
∑
i
=
1
k
λ
-
i
ρ
i
α
i
x
-
,
u
-
+
∑
j
=
1
m
μ
-
j
τ
j
β
j
x
-
,
u
-
d
2
x
-
,
u
-
,
t
-
j
=
2
μ
-
j
,
j
∈
M
.
Using the assumption∑
i
=
1
k
λ
-
i
ρ
i
/
α
i
(
x
-
,
u
-
)
+
∑
j
=
1
m
μ
-
j
τ
j
/
β
j
x
-
,
u
-
≧
0, inequality (122) implies (123)
F
x
-
,
u
-
;
η
T
x
-
,
u
-
∑
i
=
1
k
λ
-
i
∇
f
i
u
-
+
∑
j
=
1
m
t
-
j
∇
g
j
u
-
P
<
0
.
By using the constraint condition ofM
D
I
I and F
x
-
,
u
-
;
0
=
0, we obtain (124)
F
x
-
,
u
-
;
η
T
x
-
,
u
-
∑
i
=
1
k
λ
-
i
∇
f
i
u
-
+
∑
j
=
1
m
t
-
j
∇
g
j
u
-
P
=
0
,which is a contradiction to (123). Then x
-
=
u
-.
The proof of part (b) is similar to the proof of part (a).


## 6. Discussion and Conclusion

In this paper, we study the multiobjective programming problems and two kinds of dual models. Then the sufficient optimality conditions, weak dual, strong dual, and strict converse dual, results are obtained and proved under a class of new generalized invex functions assumptions for the multiobjective programming.



---

*Source: 8462602-2016-09-25.xml*