# Wolfe Type Second Order Nondifferentiable Symmetric Duality in Multiobjective Programming over Cone with Generalized (K, F)-Convexity

Authors: A. K. Tripathy
Journal: ISRN Applied Mathematics
 (2014)

---

## Abstract

A new class of second order (K, F) pseudoconvex function is introduced with example. A pair of Wolfe type second order nondifferentiable symmetric dual programs over arbitrary cones with square root term is formulated. The duality results are established under second order (K, F) pseudoconvexity assumption. Also a Wolfe type second order minimax mixed integer programming problem is formulated and the symmetric duality results are established under second order (K, F) pseudoconvexity assumption.


---

## Body

## 1. Introduction

A mathematical programming with two or more objective functions is called multiobjective programming. Often the several objectives are conflicting in nature. Pareto [1] studied multiobjective problems by reducing them to a single objective one. However, the problems were first explicitly defined and studied by Kuhn and Tucker [2]. They also proposed the definition of proper efficiency which was later modified by Geoffrion [3].In mathematical programming, a pair of primal and dual programs is called symmetric if the dual of the dual is the primal problem. The duality in linear programming is symmetric. It is not so in nonlinear programming in general. Dorn [4], Dantzig et al. [5], and Mond [6] studied symmetric duality in nonlinear programming assuming the kernel function f
(
x
,
y
) to be convex in x and concave in y. Subsequently, Mond and Weir [7] presented a distinct pair of symmetric dual nonlinear programs which admits the relaxation of the convexity/concavity assumption to pseudoconvexity/pseudoconcavity. Mond [6] initiated second order symmetric duality of Wolfe type in nonlinear programming and proved the duality theorems under second order convexity. Mangasarian [8] discussed second order duality in nonlinear programming under inclusion condition. Mond [6, page 93] and Mangasarian [8, page 609] also indicated possible computational advantages of the second order dual over the first order dual. This motivated several authors [3, 6, 9–13] in this field. Yang et al. [13] studied second order multiobjective symmetric dual programs and established the duality relations underF-convexity assumptions. Also Yang et al. [12] formulated a pair of Wolfe type second order nondifferentiable symmetric dual programs containing support function and presented the duality results underF convexity.Recently, Gulati et al. [14] studied Wolfe and Mond-Weir type second order symmetric duality over arbitrary cones and proved the duality results under generalized convexity assumption. Gulati and Geeta [15] studied Mond-Weir type second order symmetric duality in multiobjective programming over cones and established duality results under pseudoconvexity/K-F convexity assumption. Gulati and Verma [16] formulated a pair of Wolfe type nondifferentiable multiobjective symmetric duality and established the duality results under invexity assumption. Gupta and Kailey [17] formulated a pair of Wolfe type second order nondifferentiable multiobjective symmetric dual programs in which the objective function contains support function and proved the duality results under second orderF-convexity assumption. Gupta and Kailey [18] presented second order multiobjective symmetric duality involving cone-convex functions. Saini and Gulati [19] presented a pair of Wolfe type nondifferentiable second order symmetric dual programs over arbitrary cones under second order (K, F)-convexity assumption.In this paper, motivated by Saini and Gulati [19], a new class of second order (K, F) pseudoconvex/second order (K, F) strongly pseudoconvex function is introduced with example. A pair of Wolfe type second order nondifferentiable symmetric dual programs over arbitrary cone containing square root term is formulated. The duality results are established under second order (K, F) pseudoconvexity assumption.


## 2. Notation and Preliminaries

The following convention for vectors inR
n will be used:
(1)
x
<
y
⟺
x
i
<
y
i
,
i
=
1,2
,
…
,
n
,
x
≦
y
⟺
x
i
≦
y
i
,
i
=
1,2
,
…
,
n
,
x
≤
y
⟺
x
i
≤
y
i
,
i
=
1,2
,
…
,
n
,
but
x
≠
y
.Definition 1.
A setC of R
n is called a cone if, for each x
∈
C and λ
∈
R
,
λ
≥
0, we have λ
x
∈
C. Moreover, if C is convex, then it is convex cone.Definition 2.
The positive polar coneC
* of C is defined as
(2)
C
*
=
{
z
∈
R
n
∣
x
T
z
≥
0
,
∀
x
∈
C
}
.
Let C
1
⊂
R
n, C
2
⊂
R
m, and K
⊂
R
k be closed convex cones with nonempty interiors having polars C
1
*, C
2
*, and K
*, respectively. Let X
⊆
R
n and Y
⊆
R
m be open and X
×
Y
⊆
R
n
×
R
m. Let C
1
×
C
2
⊆
X
×
Y.A general multiobjective nonlinear programming problem can be expressed in the following form.Primal (P).
(3)
Minimize
f
(
x
)
=
{
f
1
(
x
)
,
f
2
(
x
)
,
…
,
f
k
(
x
)
}
Subject
to
-
g
(
x
)
∈
Q
,
x
∈
X
,
hhhhhhjjhh
where
X
⊆
R
n
is
open
,
f
:
R
n
→
R
k
,
g
:
R
n
→
R
l, Q is a closed convex cone with nonempty interior in R
m.Definition 3.
A feasible pointx
- is weakly efficient solution of (P) if there exist no other x
∈
X such that f
(
x
)
-
f
(
x
-
)
<
0.Definition 4.
A feasible pointx
- is efficient solution of (P) if there exist no other x
∈
X such that f
(
x
)
-
f
(
x
-
)
≤
0.Definition 5.
A functionF
:
X
×
X
×
R
n
→
R is sublinear in its third argument if, for all (
x
,
u
)
∈
X
×
X,(1)
F
(
x
,
u
;
a
1
+
a
2
)
≤
F
(
x
,
u
;
a
1
)
+
F
(
x
,
u
;
a
2
), for all a
1
,
a
2
∈
R
n,
(2)
F
(
x
,
u
;
α
a
)
=
α
F
(
x
,
u
;
a
), for  all α
∈
R
+.Definition 6.
Letf
:
R
n
→
R
k be thrice differentiable function. f is said to be second orderF-pseudoconvex at u
∈
X, if (
x
,
p
)
∈
X
×
R
n,
(4)
F
x
,
u
(
∇
u
f
(
u
)
+
∇
u
u
f
(
u
)
p
)
≥
0
⟹
f
(
x
)
-
f
(
u
)
+
1
2
p
T
∇
u
f
(
u
)
p
≥
0
.
Now, we are in position to give definition of second order (K, F)-pseudoconvex function and second order strongly (K, F) pseudoconvex function.Definition 7.
The thrice differentiable functionf
=
(
f
1
,
f
2
,
…
,
f
k
)
:
R
n
→
R
k is said to be second order (K, F) pseudoconvex at u
∈
X, if (
x
,
p
)
∈
X
×
R
n,
(5)
j
j
j
-
(
F
x
,
u
(
∇
u
f
k
(
u
)
+
∇
u
u
f
k
(
u
)
p
1
)
)
F
x
,
u
(
∇
u
f
1
(
u
)
+
∇
u
u
f
1
(
u
)
p
1
)
,
…
,
j
h
h
h
j
j
F
x
,
u
(
∇
u
f
k
(
u
)
+
∇
u
u
f
k
(
u
)
p
1
)
)
∉
int
⁡
K
⟹
-
(
f
1
(
x
)
-
f
1
(
u
)
+
1
2
p
1
T
∇
u
f
1
(
u
)
p
1
,
…
,
f
k
(
x
)
hhhhhhhhlh
-
f
k
(
u
)
+
1
2
p
k
T
∇
u
f
k
(
u
)
p
k
)
∉
int
⁡
K
.Definition 8.
The thrice differentiable functionf
:
R
n
→
R
k is said to be second order strongly (K, F) pseudoconvex at u
∈
X, if (
x
,
p
)
∈
X
×
R
n,
(6)
-
(
F
x
,
u
(
∇
u
f
1
(
u
)
+
∇
u
u
f
1
(
u
)
p
1
)
,
…
,
h
h
h
F
x
,
u
(
∇
u
f
k
(
u
)
+
∇
u
u
f
k
(
u
)
p
1
)
)
∉
int
⁡
K
,
⟹
(
f
1
(
x
)
-
f
1
(
u
)
+
1
2
p
1
T
∇
u
f
1
(
u
)
p
1
,
…
,
f
k
(
x
)
hhhlhh
-
f
k
(
u
)
+
1
2
p
k
T
∇
u
f
k
(
u
)
p
k
)
∈
int
⁡
K
.Definition 9.
f is second order (K, F) pseudoconcave, if -
f is second order (K, F) pseudoconvex, and f is second order strongly (K, F) pseudoconcave, if -
f is second order strongly (K, F) pseudoconvex function.Example 10.
Let(7)
K
=
{
(
x
,
y
)
∣
-
4
x
≤
y
≤
-
x
2
,
x
>
0
}
,
f
(
x
)
=
(
f
1
(
x
)
,
f
2
(
x
)
)
=
(
-
x
2
+
x
,
e
-
x
)
,
p
=
1
,
F
x
,
u
(
a
)
=
a
(
x
3
+
u
)
,
(
∇
u
f
1
(
u
)
,
∇
u
f
2
(
u
)
)
=
(
-
2
u
+
1
,
-
e
-
u
)
,
(
∇
u
u
f
1
(
u
)
,
∇
u
u
f
2
(
u
)
)
=
(
-
2
,
e
-
u
)
,
(
a
1
,
a
2
)
=
(
∇
u
f
1
(
u
)
+
∇
u
u
f
1
(
u
)
p
,
∇
u
f
2
(
u
)
+
∇
u
u
f
2
(
u
)
p
)
=
(
-
2
u
-
1,0
)
;
(
F
x
,
u
(
a
1
)
,
F
x
,
u
(
a
2
)
)
=
(
-
2
u
-
1,0
)
(
x
3
+
u
)
,
(
f
1
(
x
)
-
f
1
(
u
)
+
1
2
p
T
∇
u
u
f
1
(
u
)
p
,
f
2
(
x
)
-
f
2
(
u
)
+
1
2
p
T
∇
u
u
f
2
(
u
)
p
)
=
(
-
x
2
+
x
+
u
2
-
u
-
1
,
e
-
x
-
1
2
e
-
u
)
.
Now at
(8)
(
F
x
,
u
(
a
1
)
,
F
x
,
u
(
a
2
)
)
≥
0
⟹
x
≤
0
.
So at
(9)
(
f
1
(
x
)
-
f
1
(
u
)
+
1
2
p
T
∇
u
u
f
1
(
u
)
p
,
f
2
(
x
)
-
f
2
(
u
)
+
1
2
p
T
∇
u
u
f
2
(
u
)
p
)
≱
0
.
So f is not second orderF convex function.
Now at(10)
u
=
0
,
x
=
1
,
-
(
F
x
,
u
(
a
1
)
,
F
x
,
u
(
a
2
)
)
=
(
1,0
)
∉
int
⁡
K
,
(
f
1
(
x
)
-
f
1
(
u
)
+
1
2
p
T
∇
u
u
f
1
(
u
)
p
,
f
2
(
x
)
-
f
2
(
u
)
+
1
2
p
T
∇
u
u
f
2
(
u
)
p
)
=
(
-
1
,
e
-
1
-
1
2
)
∉
int
⁡
K
.
So f is not second order strongly (K, F) pseudoconvex. But
(11)
-
(
f
1
(
x
)
-
f
1
(
u
)
+
1
2
p
T
∇
u
u
f
1
(
u
)
p
,
f
2
(
x
)
hhhhh
-
f
2
(
u
)
+
1
2
p
T
∇
u
u
f
2
(
u
)
p
)
=
(
1
,
-
e
-
1
+
1
2
)
∉
int
⁡
K
.
So f is second order (K, F) pseudoconvex.
Now we can define second order (K, F) pseudoconvexity for a multiobjective function:
(12)
f
=
(
f
1
,
f
2
,
…
,
f
k
)
:
R
n
×
R
m
⟶
R
k
.Definition 11.
A thrice differentiable functionf
=
(
f
1
,
f
2
,
…
,
f
k
)
:
R
n
×
R
m
→
R
k is said to be second order (K, F)-pseudoconvex at u
∈
X, for fixed v, if there exists sublinear function F
:
X
×
X
×
R
n
→
R
,
q
∈
R
n
,
x
∈
X
,
v
∈
Y such that
(13)
-
(
F
x
,
u
(
∇
u
f
1
(
u
,
v
)
+
∇
u
u
f
1
(
u
,
v
)
q
1
)
,
…
,
hhh
F
x
,
u
(
∇
u
f
k
(
u
,
v
)
+
∇
u
u
f
k
(
u
,
v
)
q
k
)
)
∉
int
⁡
K
⟹
-
(
f
1
(
x
,
v
)
-
f
1
(
u
,
v
)
+
1
2
q
1
T
∇
u
f
1
(
u
,
v
)
q
1
,
…
,
hhlllhhhhh
f
k
(
x
,
v
)
-
f
k
(
u
,
v
)
+
1
2
q
k
T
∇
u
f
k
(
u
,
v
)
q
k
)
∉
int
⁡
K
.Definition 12.
A thrice differentiable functionf
=
(
f
1
,
f
2
,
…
,
f
k
)
:
R
n
×
R
m
→
R
k is said to be second order (K, G)-pseudoconvex at y
∈
Y, for fixed v, if there exists sublinear function G
:
Y
×
Y
×
R
m
→
R
,
p
∈
R
m
,
x
∈
X
,
v
∈
Y such that
(14)
-
(
G
v
,
y
(
∇
y
f
1
(
x
,
y
)
+
∇
y
y
f
1
(
x
,
y
)
p
1
)
,
…
,
h
h
h
G
v
,
y
(
∇
y
f
k
(
x
,
y
)
+
∇
y
y
f
k
(
x
,
y
)
p
k
)
)
∉
int
⁡
K
,
⟹
-
(
f
1
(
x
,
v
)
-
f
1
(
x
,
y
)
+
1
2
p
1
T
∇
y
f
1
(
x
,
y
)
p
1
,
…
,
j
j
j
j
j
j
j
j
j
j
j
j
f
k
(
x
,
v
)
-
f
k
(
x
,
y
)
+
1
2
p
k
T
∇
y
f
k
(
x
,
y
)
p
k
)
∉
int
⁡
K
.Lemma 13 (generalized Schwartz inequality).
LetB be a positive semidefinite matrix of order n. Then, for all x
,
w
∈
R
n
,
x
T
B
w
≤
(
x
T
B
x
)
1
/
2
(
w
T
B
w
)
1
/
2.
The equality holds ifB
x
=
λ
B
w for some λ
≥
0.


## 3. Wolfe Type Second Order Multiobjective Nondifferentiable Dual Programs

We consider the following pair of second order Wolfe type nondifferentiable multiobjective programming problems with k-objective.Primal (SWP). Consider
(15)
L
(
x
,
y
,
λ
,
w
,
p
)
=
Minimize
{
1
2
f
i
(
x
,
y
)
+
(
x
T
B
i
x
)
1
/
2
=
Minimizeeeeee
-
y
T
[
∇
y
f
i
(
x
,
y
)
+
∇
y
y
f
i
(
x
,
y
)
p
]
h
h
h
h
h
h
h
h
e
e
e
e
e
h
h
-
1
2
p
i
T
[
∇
y
y
f
i
(
x
,
y
)
p
i
]
,
i
=
1,2
,
…
,
k
}
l
=
Subject
to

-
∑
i
=
1
k
λ
i
[
∇
y
f
i
(
x
,
y
)
-
D
i
w
i
+
∇
y
y
f
i
(
x
,
y
)
p
i

]
hhhhhhhhhhhhhhhh
∈
C
2
*
,
(16)
w
i
T
D
i
w
i
≤
1
,
i
=
1,2
,
…
,
k
,
(17)
x
∈
C
1
,
w
i
∈
R
m
,
(18)
λ
∈
int
⁡
K
*
,
∑
i
=
1
k
λ
i
=
1
.Dual (SWD).Consider
(19)
M
(
u
,
v
,
λ
,
z
,
q
)
=
Maximize
{
1
2
f
i
(
u
,
u
)
-
(
v
T
D
i
v
)
1
/
2
=
Maximizeeeeee
-
u
T
[
∇
u
f
i
(
u
,
v
)
+
∇
u
u
f
i
(
u
,
v
)
q
i
]
=
Maximizeeeeee
-
1
2
q
i
T
[
∇
u
u
f
i
(
u
,
v
)
q
i
]
,
i
=
1,2
,
…
,
k
}
=
j
Subject
to
∑
i
=
1
k
λ
i
[
∇
u
f
i
(
u
,
v
)
+
B
i
z
i
+
∇
u
u
f
i
(
u
,
v
)
q
i
]
hhhhhhhhhhhhhh
∈
C
1
*
,
(20)
z
i
T
B
i
z
i
≤
1
,
i
=
1,2
,
…
,
k
,
(21)
v
∈
C
2
,
z
i
∈
R
n
,
(22)
λ
∈
int
⁡
K
*
,
∑
i
=
1
k
λ
i
=
1
,
where(1)
f
=
(
f
1
,
f
2
,
…
,
f
k
)
:
R
n
×
R
m
→
R
k is thrice differentiable function,
(2)
C
1 and C
2 are closed convex cones in R
n and R
m with nonempty interiors, respectively,
(3)
C
1
* and C
2
* are positive polar cones of C
1 and C
2, respectively,
(4)
K is a closed convex cone in R
k with int
⁡
K
≠
ϕ and R
+
k
⊂
K,
(5)
q
i, z
i  (
i
=
1,2
,
…
,
k
) are vectors in R
n, and p
i, w
i  (
i
=
1,2
,
…
,
k
) are vectors in R
m,
(6)
B
i and D
i  (
i
=
1,2
,
…
,
k
) are positive semidefinite matrices of order n and m, respectively.Now we establish the following theorem.Theorem 14 (weak duality theorem).
Let(
x
,
y
,
λ
,
p
) be a feasible solution for the primal (WP) and let (
u
,
v
,
λ
,
q
) be a feasible solution for the dual (WD). Suppose there exist sublinear functional F
:
X
×
X
×
R
n
→
R and G
:
Y
×
Y
×
R
m
→
R satisfying (1)
F
x
,
u
(
a
)
-
u
T
a
≥
0,  for  all  (
x
,
u
)
∈
C
1
×
C
1
,
a
∈
C
1
*,
(2)
G
v
,
y
(
b
)
-
y
T
b
≥
0,  for  all  (
v
,
y
)
∈
C
2
×
C
2
,
b
∈
C
2
*.Furthermore assume that, for each i, f
i
(
·
,
v
)
+
(
·
)
T
B
i
z
i is second order (K, F)-pseudoconvex at u for fixed v and f
i
(
x
,
·
)
+
(
·
)
T
D
i
w
i is second order pseudoconcave at y for fixed x:
(23)
Inf
⁡
(
WP
)
-
Sup
⁡
(
WD
)
∈
int
⁡
K
.Proof.
Since(
u
,
v
,
λ
,
q
) is feasible solution for (WD), from dual constraint (5) we have
(24)
a
=
∑
i
=
1
k
λ
i
[
∇
u
f
i
(
u
,
v
)
+
B
i
z
i
+
∇
u
u
f
i
(
u
,
v
)
q
i
]
∈
C
1
*
.
So
(25)
u
∈
C
1
⟹
u
T
a
=
u
T
(
∑
i
=
1
k
λ
i
[
∇
u
f
i
(
u
,
v
)
+
B
i
z
i
+
∇
u
u
f
i
(
u
,
v
)
q
i
]
)
≥
0
.
Again hypothesis (1) implies F
x
,
u
(
a
)
≥
0. Consider
(26)
⟹
F
x
,
u
(
∑
i
=
1
k
λ
i
[
∇
u
f
i
(
u
,
v
)
+
B
i
z
i
+
∇
u
u
f
i
(
u
,
v
)
q
i
]
)
≥
0
.
Since F is sublinear with respect to third argument,
(27)
⟹
∑
i
=
1
k
λ
i
F
x
,
u
(
[
∇
u
f
i
(
u
,
v
)
+
B
i
z
i
+
∇
u
u
f
i
(
u
,
v
)
q
i
]
)
≥
0
.
Since λ
∈
int
⁡
K, the above inequality can be written as
(28)
-
(
F
x
,
u
(
[
∇
u
f
1
(
u
,
v
)
+
B
1
z
1
+
∇
u
u
f
1
(
u
,
v
)
q
1
]
)
,
…
,
h
h
h
F
x
,
u
(
[
∇
u
f
k
(
u
,
v
)
+
B
k
z
k
+
∇
u
u
f
k
(
u
,
v
)
q
k
]
)
)
∉
int
⁡
K
.
So second order (K, F)-pseudoconvexity of f
i
(
·
,
v
)
+
(
·
)
T
B
i
z
i at u for fixed v implies that
(29)
-
(
f
1
(
x
,
v
)
+
(
x
)
T
B
1
z
1
-
f
1
(
u
,
v
)
-
(
u
)
T
B
1
z
1
+
1
2
q
1
T
∇
u
u
f
1
(
u
,
v
)
q
1
,
.
.
.
,
f
k
(
x
,
v
)
+
(
x
)
T
B
k
z
k
-
f
k
(
u
,
v
)
-
(
u
)
T
B
k
z
k
+
1
2
q
k
T
∇
u
u
f
k
(
u
,
v
)
q
k
)
∉
int
⁡
K
.
This implies that, for λ
∈
int
⁡
K,
(30)
-
∑
i
=
1
k
λ
i
[
1
2
f
i
(
x
,
v
)
+
(
x
)
T
B
i
z
i
-
f
i
(
u
,
v
)
hhhhhhh
-
(
u
)
T
B
i
z
i
-
1
2
q
i
T
∇
u
u
f
i
(
u
,
v
)
q
i
]
≱
0
⟹
∑
i
=
1
k
λ
i
[
1
2
f
i
(
x
,
v
)
+
(
x
)
T
B
i
z
i
-
f
i
(
u
,
v
)
hhhhhhhhh
-
(
u
)
T
B
i
z
i
-
1
2
q
i
T
∇
u
u
f
i
(
u
,
v
)
q
i
]
≥
0
.
Similarly (
u
,
v
,
λ
,
q
) is feasible solution for (WD), so from primal constraint (1) we have
(31)
b
=
-
∑
i
=
1
k
λ
i
[
∇
y
f
i
(
x
,
y
)
-
D
i
w
i
+
∇
y
y
f
i
(
x
,
y
)
p
i
]
∈
C
2
*
.
So
(32)
y
∈
C
2
⟹
y
T
b
=
-
y
T
(
∑
i
=
1
k
λ
i
[
∇
y
f
i
(
x
,
y
)
hhhhhhhhhlh
-
D
i
w
i
+
∇
y
y
f
i
(
x
,
y
)
p
i
]
∑
i
=
1
k
λ
i
[
∇
y
f
i
(
x
,
y
)
)
≥
0
.
Again hypothesis (2) impliesG
v
,
y
(
b
)
≥
0. Consider
(33)
⟹
G
v
,
y
(
-
∑
i
=
1
k
λ
i
[
∇
y
f
i
(
x
,
y
)
-
D
i
w
i
+
∇
y
y
f
i
(
x
,
y
)
p
i
]
)
≥
0
.
Since G is sublinear with respect to third argument,
(34)
⟹
∑
i
=
1
k
λ
i
G
v
,
y
(
-
[
∇
y
f
i
(
x
,
y
)
-
D
i
w
i
+
∇
y
y
f
i
(
x
,
y
)
p
i
]
)
≥
0
.
Since λ
∈
int
⁡
K, the above inequality can be written as
(35)
⟹
-
(
G
v
,
y
(
-
[
∇
y
f
1
(
x
,
y
)
-
D
1
w
1
+
∇
y
y
f
1
(
x
,
y
)
p
1
]
)
,
…
…
,
G
v
,
y
(
-
[
∇
y
f
k
(
x
,
y
)
-
D
k
w
k
+
∇
y
y
f
k
(
x
,
y
)
p
k
]
)
)
∉
int
⁡
K
.
So second order (K, F)-pseudoconcavity of f
i
(
x
,
·
)
-
(
·
)
T
D
i
w
i at y for fixed x implies that
(36)
-
(
-
[
1
2
f
1
(
x
,
v
)
-
(
v
)
T
D
1
w
1
-
f
1
(
x
,
y
)
+
(
y
)
T
D
1
w
1
h
h
h
h
h
h
+
1
2
p
1
T
∇
y
y
f
1
(
x
,
y
)
p
1
]
,
…
,
h
h
h
-
[
1
2
f
k
(
x
,
v
)
-
(
v
)
T
D
k
z
k
-
f
k
(
x
,
y
)
+
(
y
)
T
D
k
w
k
h
h
h
h
h
h
+
1
2
p
k
T
∇
y
y
f
k
(
x
,
y
)
p
k
]
)
hh
∉
int
⁡
K
.
This implies that, for λ
∈
int
⁡
K,
(37)
-
∑
i
=
1
k
λ
i
[
-
[
1
2
f
k
(
x
,
v
)
-
(
v
)
T
D
k
w
k
-
f
k
(
x
,
y
)
hhhhhhhhh
+
(
y
)
T
D
k
w
k
+
1
2
p
k
T
∇
y
y
f
k
(
x
,
y
)
p
k
]
]
≱
0
⟹
∑
i
=
1
k
λ
i
[
1
2
-
f
i
(
x
,
v
)
+
(
v
)
T
D
i
w
i
+
f
i
(
x
,
y
)
hhhhhhhh
-
(
y
)
T
D
i
w
i
-
1
2
p
i
T
∇
y
y
f
i
(
x
,
y
)
p
i
]
≥
0
.
Adding (30) and (37), we get
(38)
⟹
∑
i
=
1
k
λ
i
[
1
2
f
i
(
x
,
y
)
+
(
x
)
T
B
i
z
i
-
(
y
)
T
D
i
w
i
hhhhhhhh
-
1
2
p
i
T
∇
y
y
f
i
(
x
,
y
)
p
i
]
-
∑
i
=
1
k
λ
i
[
1
2
f
i
(
u
,
v
)
-
(
v
)
T
D
i
w
i
+
u
T
B
i
z
i
hhhhhhhhhhl
-
1
2
q
i
T
∇
u
u
f
i
(
u
,
v
)
q
i
]
≥
0
.
Now from Schwartz inequality (Lemma 13), (17), and (21), we have
(39)
x
T
B
i
z
i
≤
(
x
T
B
i
x
)
1
/
2
(
z
i
T
B
i
z
i
)
1
/
2
≤
(
x
T
B
i
x
)
1
/
2
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
i
=
1,2
,
…
,
k
,
v
T
D
i
w
i
≤
(
v
T
D
i
v
)
1
/
2
(
w
i
T
D
i
w
i
)
1
/
2
≤
(
v
T
D
i
v
)
1
/
2
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
l
i
=
1,2
,
…
,
k
.
Also from primal constraint (15), we have
(40)
∑
i
=
1
k
λ
i
[
∇
y
f
i
(
x
,
y
)
-
D
i
w
i
+
∇
y
y
f
i
(
x
,
y
)
p
i
]
∈
C
2
*
.
For y
∈
C
1, y
T
(
∑
i
=
1
k
λ
i
[
∇
y
f
i
(
x
,
y
)
-
D
i
w
i
+
∇
y
y
f
i
(
x
,
y
)
p
i
]
)
≥
0
(41)
-
∑
i
=
1
k
λ
i
[
y
T
D
i
w
i
]
≥
-
y
T
(
∑
i
=
1
k
λ
i
[
∇
y
f
i
(
x
,
y
)
+
∇
y
y
f
i
(
x
,
y
)
p
i
]
)
.
Similarly from dual constraint (19), we have
(42)
⟹
-
∑
i
=
1
k
λ
i
[
u
T
B
i
z
i
]
≥
-
u
T
(
∑
i
=
1
k
λ
i
[
∇
u
f
i
(
u
,
v
)
+
∇
u
u
f
i
(
u
,
v
)
q
i
]
)
.
Using (39), (41), and (42) in (38), we obtain that
(43)
∑
i
=
1
k
λ
i
{
f
i
(
x
,
y
)
+
(
x
T
B
i
x
)
1
/
2
hhhhhl
-
y
T
[
∇
y
f
i
(
x
,
y
)
+
∇
y
y
f
i
(
x
,
y
)
p
]
hhhhhl
-
1
2
p
i
T
[
∇
y
y
f
i
(
x
,
y
)
p
i
]
}
-
∑
i
=
1
k
λ
i
{
1
2
f
i
(
u
,
u
)
-
(
v
T
D
i
v
)
1
/
2
hhhhhhhhh
-
u
T
[
∇
u
f
i
(
u
,
v
)
+
∇
u
u
f
i
(
u
,
v
)
q
i
]
hhhhhhhhh
-
1
2
q
i
T
[
∇
u
u
f
i
(
u
,
v
)
q
i
]
}
≥
0
⟹
Inf
⁡
⁡
(
WP
)
-
Sup
⁡
⁡
(
WD
)
∈
int
⁡
⁡
K
.Theorem 15 (strong duality).
Let(
x
-
,
y
-
,
λ
-
,
w
-
,
p
-
) be weakly efficient solution of (WP) such that(i)
∇
y
y
f
i
(
x
-
,
y
-
) is nonsingular,
(ii)
the matrix∑
i
=
1
k
λ
i
[
∇
y
(
∇
y
y
f
i
(
x
-
,
y
-
)
p
-
)
] is positive definite,
(iii)
the set{
∇
y
f
1
-
C
1
w
1
,
…
,
∇
y
f
k
-
C
k
w
k
} is linearly independent,
(iv)
∑
i
=
1
k
λ
-
i
(
∇
y
(
∇
y
y
f
i
p
-
)
)
p
-
∉
span
⁡
{
∇
y
f
1
-
C
1
w
1
,
…
,
∇
y
f
k
-
C
k
w
1
}
∖
{
0
}.Then there exist z
-
i
∈
R
n such that (
x
-
,
y
-
,
λ
-
,
z
-
,
q
-
=
0
) is feasible for (WD) and two objective values of (WP) and (WD) are equal. Also, if the hypotheses of Theorem 14 are satisfied for all feasible solution of (WP) and (WD), then (
x
-
,
y
-
,
λ
-
,
z
-
,
q
-
=
0
) is an efficient solution of (WD).Proof.
Since(
x
-
,
y
-
,
λ
-
,
w
-
,
p
-
) is weakly efficient solution of (WD), by the Fritz-John necessary optimality condition on convex cone domain given in Bazaraa and Goode [20], there exist α
∈
K
*
,
β
∈
C
2
,
γ
,
τ
i
∈
R
+ such that the following conditions are satisfied at (
x
-
,
y
-
,
λ
-
,
w
-
,
p
-
):
(44)
(
x
-
x
-
)
T
×
(
∑
i
=
1
k
α
i
[
∇
x
f
i
+
B
i
z
i
]
+
∑
i
=
1
k
λ
i
(
∇
y
x
f
i
)
T
(
β
-
γ
y
-
)
H
H
H
H
+
∑
i
=
1
k
λ
i
(
∇
x
(
∇
y
y
f
i
p
-
i
)
)
T
[
β
-
γ
y
-
-
α
i
p
-
i
]
)
≥
0
,
∀
x
∈
C
1
,
(45)
∑
i
=
1
k
(
α
i
-
γ
λ
i
)
(
∇
y
f
i
-
C
i
w
i
)
+
∑
i
=
1
k
λ
i
(
∇
y
y
f
i
)
T
[
β
-
γ
(
y
-
+
p
-
)
]
+
∑
i
=
1
k
λ
i
(
∇
y
(
∇
y
y
f
i
p
-
i
)
)
T
(
β
-
γ
y
-
-
1
2
γ
p
-
i
)
=
0
,
(46)
∑
i
=
1
k
λ
i
(
∇
y
y
f
i
)
T
(
β
-
γ
y
-
-
γ
p
-
i
)
=
0
,
(47)
∑
i
=
1
k
λ
i
(
∇
y
f
i
-
C
i
w
i
+
∇
y
y
f
i
p
-
i
)
T
β
=
0
,
(48)
x
T
B
i
z
i
=
(
x
T
B
i
x
)
1
/
2
,
i
=
1,2
,
…
,
k
,
(49)
λ
-
=
(
λ
-
1
,
λ
-
2
,
…
,
λ
-
k
)
∈
int
⁡
K
,
(50)
α
i
C
i
y
-
+
(
β
-
λ
y
-
)
λ
i
C
i
=
2
τ
i
C
i
w
i
,
(51)
τ
i
(
w
-
i
T
C
i
w
-
i
-
1
)
=
0
,
(52)
z
-
i
T
B
i
z
-
i
≤
1
,
(53)
(
α
,
β
,
γ
,
τ
)
≥
0
,
(54)
(
α
,
β
,
γ
,
τ
)
≠
0
.
Since ∇
y
y
f
i is nonsingular, (46) implies that β
=
γ
(
y
-
+
p
-
i
)
(55)
β
=
γ
(
y
-
+
p
-
i
)
.
We claim that γ
>
0. Indeed if γ
=
0, then (55) implies β
=
0, which contradicts (54).
Hence(56)
γ
>
0
.
Since γ
>
0, using (55) in (45), we get
(57)
∑
i
=
1
k
λ
i
[
∇
y
(
∇
y
y
f
i
p
-
)
]
T
p
-
i
=
-
2
γ
∑
i
=
1
k
(
α
i
-
γ
λ
i
)
[
∇
y
f
i
-
C
i
w
i
]
,
which by hypothesis (ii) and (iv) yields
(58)
p
-
i
=
0
,
i
=
1,2
,
…
,
k
.
From (55) and (58), we obtain
(59)
β
=
γ
y
-
.
Using (58) and (59) and hypothesis (iii) in (45), we get
(60)
α
=
γ
λ
.
Again using (58), (59), and (60) in (44), we get
(61)
(
x
-
x
-
)
T
∑
i
=
1
k
λ
i
[
∇
x
f
i
+
B
i
z
i
]
≥
0
,
∀
x
∈
C
1
.
Let x
∈
C
1. Then x
+
x
-
∈
C
1 and so (61) implies
(62)
(
x
)
T
∑
i
=
1
k
λ
i
[
∇
x
f
i
+
B
i
z
i
]
≥
0
,
∀
x
∈
C
1
⟹
∑
i
=
1
k
λ
i
[
∇
x
f
i
+
B
i
z
i
]
∈
C
1
*
.
Also from (56), (59), and β
∈
C
2, we obtain
(63)
y
-
∈
C
2
.
Thus, from (52), (62), and (63), we obtain that (
x
-
,
y
-
,
z
-
,
λ
-
,
q
-
=
0
) satisfies the dual constraints (19), (20), (21), and (22).
Thus(
x
-
,
y
-
,
z
-
,
λ
-
,
q
-
=
0
) is feasible for (WD).
Let(
2
τ
i
/
α
i
)
=
t; then t
≥
0. From (50) and (59), we get
(64)
C
i
y
-
=
t
C
i
w
-
i
.
This is a condition of Schwartz inequality:
(65)
y
-
C
i
w
-
i
=
(
y
-
T
C
i
w
-
i
)
1
/
2
(
w
-
i
T
C
i
w
-
i
)
.
In case τ
i
>
0, from (51) we get w
-
i
T
C
i
w
-
=
1. So (65) implies y
-
C
i
w
-
i
=
(
y
-
T
C
i
w
-
i
)
1
/
2.
In caseτ
i
=
0, we get t
=
0. So C
i
y
-
=
0. Hence y
-
C
i
w
-
i
=
(
y
-
T
C
i
w
-
i
)
1
/
2. Thus in either case
(66)
y
-
C
i
w
-
i
=
(
y
-
T
C
i
w
-
i
)
1
/
2
.
So using (48) and (66), we obtained that the two objective values are equal; that is,
(67)
L
(
x
-
,
y
-
,
w
-
,
λ
-
,
p
-
=
0
)
=
M
(
x
-
,
y
-
,
z
-
,
λ
-
,
q
-
=
0
)
.
Now we claim that (
x
-
,
y
-
,
z
-
,
λ
-
,
q
-
=
0
) is an efficient solution of (WD). If this would not be the case, then there would exist a feasible solution (
u
-
,
v
-
,
z
-
,
λ
-
,
q
-
=
0
) such that
(68)
M
(
x
-
,
y
-
,
z
-
,
λ
-
,
q
-
=
0
)
≤
M
(
u
-
,
v
-
,
z
-
,
λ
-
,
q
-
=
0
)
⟹
L
(
x
-
,
y
-
,
z
-
,
λ
-
,
q
-
=
0
)
≤
M
(
u
-
,
v
-
,
z
-
,
λ
-
,
q
-
=
0
)
.
This is a contradiction to weak duality Theorem 14.
Hence(
x
-
,
y
-
,
z
-
,
λ
-
,
q
-
=
0
) is efficient solution.Theorem 16 (converse duality theorem).
Let(
x
-
,
y
-
,
λ
-
,
w
-
,
p
-
) be a weakly efficient solution of (WP) such that (i)
∇
u
u
f
i
(
u
-
,
v
-
) is nonsingular,
(ii)
the matrix∑
i
=
1
k
λ
i
[
∇
u
(
∇
u
u
f
i
(
u
-
,
v
-
)
q
-
)
] is positive definite,
(iii)
the set{
∇
u
f
1
+
B
1
z
1
,
…
,
∇
u
f
k
+
B
k
z
k
} is linearly independent,
(iv)
∑
i
=
1
k
λ
-
i
(
∇
u
(
∇
u
u
f
i
q
-
)
)
q
-
∉
span
⁡
{
∇
u
f
1
+
B
1
z
1
,
…
,
∇
u
f
k
+
B
k
z
k
}
∖
{
0
}.Then there exist w
-
i
∈
R
m such that (
u
-
,
v
-
,
λ
-
,
w
-
,
p
-
=
0
) is feasible for (WD) and two objective values of (WP) and (WD) are equal. Also, if the hypotheses of Theorem 14 are satisfied for all feasible solution of (WP) and (WD), then (
u
-
,
v
-
,
λ
-
,
w
-
,
p
-
=
0
) is an efficient solution of (WD).Proof.
The proof follows on lines of Theorem15.


## 4. Wolfe Type Minimax Mixed Integer Programming

LetU and V be two arbitrary sets of integers in R
n
1
(
0
≤
n
1
≤
n
) and R
m
1
(
0
≤
m
1
≤
m
), respectively. Throughout this section, we constrained some of the components of the vector variables x
∈
R
n and y
∈
R
m to belong to arbitrary sets of integers U and V, respectively. Then we write (
x
,
y
)
=
(
x
1
,
x
2
,
y
1
,
y
2
), where x
1
=
(
x
1
,
x
2
,
…
,
x
n
) and y
1
=
(
y
1
,
y
2
,
…
,
y
n
). x
2 and y
2 are the vectors of the remaining components of x and y, respectively.Definition 17.
Lets
1
,
s
2
,
…
,
s
r be elements of an arbitrary vector space. A vector function θ
(
s
1
,
s
2
,
…
,
s
r
) will be called additively separable with respect to s
1, if there exist vector function θ
1
(
s
1
) (independent of s
2
,
…
,
s
r ) and θ
2
(
s
2
,
…
,
s
r
) (independent of s
1) such that θ
(
s
1
,
s
2
,
…
,
s
r
)
=
θ
1
(
s
1
)
+
θ
2
(
s
2
,
…
,
s
r
).
We consider the following pair of Wolfe type nondifferentiable minimax mixed integer symmetric primal and dual programs:Primal (WIP).Consider
(69)
max
⁡
x
1
min
⁡
x
2
,
y
,
w
,
p
{
f
i
(
x
,
y
)
+
(
(
x
2
)
T
B
i
x
)
1
/
2
max
⁡
x
1
min
⁡
x
2
,
y
,
w
,
p
hlhh
-
(
y
2
)
T
[
∇
y
2
f
i
(
x
,
y
)
+
∇
y
2
y
2
f
i
(
x
,
y
)
p
]
max
⁡
x
1
min
⁡
x
2
,
y
,
w
,
p
hhh
-
1
2
p
i
T
[
∇
y
2
y
2
f
i
(
x
,
y
)
p
i
]
,
i
=
1,2
,
…
,
k
}
Subject
to
-
∑
i
=
1
k
λ
i
[
∇
y
2
f
i
(
x
,
y
)
-
D
i
w
i
+
∇
y
2
y
2
f
i
(
x
,
y
)
p
i
]
HHHHHhhHHHHH
∈
C
2
*
,
w
i
T
D
i
w
i
≤
1
,
i
=
1,2
,
…
,
k
,
x
2
∈
C
1
,
x
1
∈
U
,
y
1
∈
V
,
w
i
∈
R
m
,
λ
∈
int
⁡
K
*
,
∑
i
=
1
k
λ
i
=
1
.Dual (WID). Consider
(70)
min
⁡
v
1
⁡
max
⁡
u
,
v
2
,
z
⁡
{
1
2
f
i
(
u
,
v
)
-
(
(
v
2
)
T
D
i
v
)
1
/
2
hhhhhhhhlh
+
(
u
2
)
T
[
∇
u
2
f
i
(
u
,
v
)
+
∇
u
2
u
2
f
i
(
u
,
v
)
q
i
]
hhhhhhhhh
-
1
2
q
i
T
[
∇
u
2
u
2
f
i
(
u
,
v
)
q
i
]
,
i
=
1,2
,
…
,
k
}
Subject
to
∑
i
=
1
k
λ
i
[
∇
u
2
f
i
(
u
,
v
)
+
B
i
z
i
+
∇
u
2
u
2
f
i
(
u
,
v
)
q
i
]
hhhhhhhhhhhh
∈
C
1
*
,
z
i
T
B
i
z
i
≤
1
,
i
=
1,2
,
…
,
k
,
v
2
∈
C
2
,
u
1
∈
U
,
v
1
∈
V
,
z
i
∈
R
n
,
λ
∈
int
⁡
K
*
,
∑
i
=
1
k
λ
i
=
1
,
where(1)
f
=
(
f
1
,
f
2
,
…
,
f
k
)
:
R
n
×
R
m
→
R
k is thrice differentiable function,
(2)
C
1 and C
2 are closed convex cones in R
n
-
n
1 and R
m
-
m
1 with nonempty interiors, respectively,
(3)
C
1
* and C
2
* are positive polar cones of C
1 and C
2, respectively,
(4)
K is a closed convex cone in R
k with int
⁡
K
≠
ϕ and R
+
k
⊂
K,
(5)
q
i
,
z
i
(
i
=
1,2
,
…
,
k
) are vectors in R
n, and p
i
,
w
i
(
i
=
1,2
,
…
,
k
) are vectors in R
m,
(6)
B
i and D
i
(
i
=
1,2
,
…
,
k
) are positive semidefinite matrices of order n
-
n
1 and m
-
m
1, respectively.Theorem 18 (symmetric duality).
Let(
x
-
,
y
-
,
λ
-
,
w
-
,
p
-
) be a weakly efficient solution of (WIP). Also (i)
f
(
x
,
y
) is additively separable with respect to x
1 or y
1; that is, f
i
(
x
1
,
x
2
,
y
) = f
i
1
(
x
1
)
+
f
i
2
(
x
2
,
y
),
(ii)
f
(
x
,
y
) is thrice differentiable in x
2 and y
2,
(iii)
∇
y
2
y
2
f
(
x
-
,
y
-
) is nonsingular,
(iv)
the vectorp
-
T
∇
y
2
(
∇
y
2
y
2
f
(
x
-
,
y
-
)
p
-
)
=
0
⇒
p
-
=
0.Furthermore, for any feasible solution (
x
,
y
,
λ
,
w
,
p
) in (WIP) and for any feasible solution (
u
,
v
,
λ
,
z
,
q
) in (WID), suppose there exist functional F
′
:
X
′
×
X
′
×
R
n
-
n
1
→
R and G
′
:
Y
′
×
Y
′
×
R
m
-
m
1
→
R such that
(v)
(
f
12
(
u
2
,
v
)
+
(
u
2
)
T
B
1
z
1
,
…
,
f
k
2
(
u
2
,
v
)
+
(
u
2
)
T
B
k
z
k
) is second order (
K
,
F
′
) pseudoconvex at u
2 with respect to q
∈
R
n
-
n
1 for each (
u
1
,
v
) and (
f
12
(
x
,
y
2
)
-
(
y
2
)
T
D
1
w
1
,
…, f
12
(
x
,
y
2
)
-
(
y
2
)
T
D
1
w
1
) is second order (
K
,
G
′
) pseudoconcave at y
2 with respect to p
∈
R
m
-
m
1 for each (
x
,
y
1
),
(vi)
F
x
2
,
u
2
′
(
a
)
-
(
u
2
)
T
a
≥
0, for all a
∈
C
1
*, and G
v
2
,
y
2
′
(
b
)
-
(
y
2
)
T
b
≥
0, for  all b
∈
C
2
*.
Then there existz
∈
R
n
-
n
1 such that (
x
-
,
y
-
,
λ
-
,
z
-
,
q
-
=
0
) is efficient solution for dual and optimal values (WIP) and (WID) are equal.Proof.
Let(71)
s
=
max
⁡
x
1
⁡
min
⁡
x
2
,
y
,
w
⁡
{
f
i
(
x
,
y
)
+
(
(
x
2
)
T
B
i
x
2
)
1
/
2
-
(
y
2
)
T
s
=
ma
x
x
1
mi
n
x
2
,
,
w
×
[
∇
y
2
f
i
(
x
,
y
)
+
∇
y
2
y
2
f
i
(
x
,
y
)
p
i
]
s
=
ma
x
x
1
i
n
x
2
l
l
,
y
,
w
-
1
2
p
i
T
∇
y
2
y
2
f
i
(
x
,
y
)
p
i
,
s
=
ma
x
x
1
mi
n
x
2
,
y
,
w
i
=
1,2
,
…
,
k
;
(
x
,
y
,
w
,
p
)
∈
S
(
(
x
2
)
T
B
i
x
2
)
1
/
2
}
,
t
=
min
⁡
v
1
⁡
max
⁡
u
,
v
2
,
z
⁡
{
f
i
(
u
,
v
)
-
(
(
v
2
)
T
C
i
v
2
)
1
/
2
t
h
=
mi
n
v
1
lx
u
,
v
2
,
z
+
(
u
2
)
T
[
∇
u
2
f
i
(
u
,
v
)
+
∇
u
2
u
2
f
i
(
u
,
v
)
q
i

]
t
=
mi
n
v
1
h
lx
u
,
v
2
,
z
-
1
2
q
i
T
∇
u
2
u
2
f
i
(
u
,
v
)
q
i
,
t
=
mi
n
v
1
h
lx
u
,
v
2
,
z
i
=
1,2
,
…
,
k
;
(
u
,
v
,
z
,
q
)
∈
T
(
(
x
2
)
T
B
i
x
2
)
1
/
2
}
,
where S and T are a feasible region of primal (WIP) and dual (WID), respectively.
Sincef
i
(
x
,
y
) is additively separable with respect to x
1 or y
1 (say with respect to x
1) from definition, it follows that f
i
(
x
,
y
)
=
f
i
1
(
x
1
)
+
f
i
2
(
x
2
,
y
).
Therefore(72)
∇
y
2
f
i
(
x
,
y
)
=
∇
y
2
f
i
2
(
x
2
,
y
)
,
∇
y
2
y
2
f
i
(
x
,
y
)
=
∇
y
2
y
2
f
i
2
(
x
2
,
y
)
.
So the primal (WIP) can be written as(73)
s
=
max
⁡
x
1
⁡
min
⁡
x
2
,
y
,
w
⁡
{
f
i
1
(
x
1
)
+
f
i
2
(
x
2
,
y
)
(
(
x
2
)
T
B
i
x
2
)
1
/
2
s
=
ma
x
x
1
m
x
2
,
y
,
w
+
(
(
x
2
)
T
B
i
x
2
)
1
/
2
-
(
y
2
)
T
s
=
ma
x
x
1
m
x
2
,
y
,
w
×
[
∇
y
2
f
i
2
(
x
,
y
)
+
∇
y
2
y
2
f
i
2
(
x
,
y
)
p
i
]
s
=
ma
x
x
1
m
x
2
,
y
,
w
-
1
2
p
i
T
∇
y
2
y
2
f
i
2
(
x
2
,
y
)
p
i
,
s
=
.
.
ma
x
x
1
m
x
2
,
y
,
w
(
(
x
2
)
T
B
i
x
2
)
1
/
2
i
=
1,2
,
…
,
k
;
(
x
,
y
,
w
,
p
)
∈
S
}
or
(74)
s
=
max
⁡
x
1
⁡
min
⁡
y
1
⁡
{
f
i
1
(
x
1
)
+
φ
i
(
y
1
)
;
x
1
∈
U
,
y
1
∈
V
}
,
where (WIP0):
(75)
φ
i
(
y
1
)
=
min
⁡
x
2
,
y
2
,
w
⁡
{
f
i
2
(
x
2
,
y
)
+
(
(
x
2
)
T
B
i
x
2
)
1
/
2
-
(
y
2
)
T
HHHhhHhh
×
[
∇
y
2
f
i
2
(
x
,
y
)
+
∇
y
2
y
2
f
i
2
(
x
,
y
)
p
i
]
HHHhhHhh
-
1
2
p
i
T
∇
y
2
y
2
f
i
2
(
x
2
,
y
)
p
i
{
+
(
(
x
2
)
T
B
i
x
2
)
1
/
2
-
(
y
2
)
T
}
=
Subject
to
-
∑
i
=
1
k
λ
i
[
∇
y
2
f
i
2
(
x
2
,
y
)
-
C
i
w
i
hhhhhhhhhhhhhhhhhhhlh
+
∇
y
2
y
2
f
i
2
(
x
2
,
y
)
p
i
]
∈
C
2
*
,
(
w
i
)
T
C
i
w
i
≤
1
,
i
=
1,2
,
…
,
k
,
λ
∈
int
⁡
K
*
,
∑
i
=
1
k
λ
i
=
1
,
x
2
∈
C
1
,
w
i
∈
R
m
-
m
1
,
p
i
∈
R
m
-
m
1
.
Similarly the dual (WID) can be written as
(76)
t
=
min
⁡
v
1
⁡
max
⁡
u
1
⁡
{
f
i
1
(
u
1
)
+
θ
(
v
1
)
,
u
1
∈
U
,
v
1
∈
V
}
,
where (WID0):
(77)
θ
i
(
v
1
)
=
max
⁡
u
2
,
v
2
,
z
,
q
⁡
{
1
2
f
i
2
(
u
2
,
v
)
ma
x
u
2
,
v
2
,
z
,
q
-
(
(
v
2
)
T
C
i
v
2
)
1
/
2
-
(
u
2
)
T
ma
x
u
2
,
v
2
,
z
,
q
×
[
∇
u
2
f
i
2
(
u
,
v
)
+
∇
u
2
u
2
f
i
2
(
u
,
v
)
q
i
]
ma
x
u
2
,
v
2
,
z
,
q
-
1
2
q
i
T
∇
u
2
u
2
f
i
2
(
u
2
,
v
)
q
i
}
Subject
to
∑
i
=
1
k
λ
i
[
∇
u
2
f
i
2
(
u
2
,
v
)
+
B
i
z
i
hjjjjjjjjjjjjjjjhhhhhhhlh
+
∇
u
2
u
2
f
i
2
(
u
2
,
v
)
q
i
]
∈
C
1
*
,
(
z
i
)
T
B
i
z
i
≤
1
,
i
=
1,2
,
…
,
k
,
λ
∈
int
⁡
K
*
,
∑
i
=
1
k
λ
i
=
1
,
v
2
∈
C
2
,
z
i
∈
R
n
-
n
1
,
q
i
∈
R
n
-
n
1
.
For any given y
1 and v
1, programs (WIP0) and (WID0) are a pair of Wolfe type second order nondifferentiable multiobjective symmetric dual programs studied in Section 3 and hence in view of hypothesis (ii)–(vi), Theorems 14 and 15 become applicable. Therefore y
1
=
y
-
1
=
v
1, and we obtain q
-
=
0
,
φ
(
y
1
)
=
θ
(
y
-
1
). So the two optimal values are equal and (
x
-
,
y
-
,
λ
-
,
z
-
,
q
-
=
0
) is an efficient solution for the dual.


## 5. Special Cases

(i) IfB
i
=
D
i
=
0, k
=
1, then the problems (SWP) and (SWD) can be reduced to the problem proposed by Gulati et al. [14] as follows.Primal (WP). 						(78)
Minimize
f
(
x
,
y
)
-
y
T
[
∇
y
f
(
x
,
y
)
+
∇
y
y
f
(
x
,
y
)
p
]
Mingggggggggggimize
-
1
2
p
T
[
∇
y
y
f
(
x
,
y
)
p
]
Subject
to
-
[
∇
y
f
i
(
x
,
y
)
+
∇
y
y
f
i
(
x
,
y
)
p
i
]
∈
C
2
*
,
Subject
to
hhhhhhhhhhhhhhhhhhhhhhh
x
∈
C
1
.Dual (WD). 						(79)
Maximize
f
(
u
,
u
)
+
u
T
[
∇
u
f
(
u
,
v
)
+
∇
u
u
f
(
u
,
v
)
q
]
Maximize
-
1
2
q
T
[
∇
u
u
f
(
u
,
v
)
q
]
Subject
to
∇
u
f
(
u
,
v
)
+
∇
u
u
f
(
u
,
v
)
q
∈
C
1
*
,
h
h
h
h
h
h
h
h
h
h
h
h
l
l
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
v
∈
C
2
.(ii) Ifk
=
1, C
1
=
R
+
n, C
2
=
R
+
m, (
x
T
B
x
)
1
/
2
=
s
(
x
∣
C
′
), and (
y
T
D
y
)
1
/
2
=
s
(
y
∣
D
′
), where C
′
=
{
B
x
∣
x
T
B
x
≤
1
}, D
′
=
{
D
y
∣
y
T
D
y
≤
1
}, then the problems (SWP) and (SWD) can be reduced to the problem proposed by Yang et al. [12].Primal (WP).
(80)
Minimize
f
(
x
,
y
)
+
s
(
x
∣
C
′
)
Minimize
-
y
T
[
∇
y
f
(
x
,
y
)
+
∇
y
y
f
(
x
,
y
)
p
]
Minimize
-
1
2
p
T
[
∇
y
y
f
(
x
,
y
)
p
]
Subject
to
∇
y
f
(
x
,
y
)
-
z
+
∇
y
y
f
(
x
,
y
)
p
≤
0
,
H
H
H
H
H
H
H
h
x
≥
0
,
z
∈
D
′
.Dual (WD). 						(81)
Maximize
f
(
u
,
u
)
-
s
(
v
∣
D
′
)
Maximize
-
u
T
[
∇
u
f
(
u
,
v
)
+
∇
u
u
f
(
u
,
v
)
q
]
Minimize
-
1
2
q
T
[
∇
u
u
f
(
u
,
v
)
q
]
Subject
to
∇
u
f
(
u
,
v
)
+
∇
u
u
f
(
u
,
v
)
q
≥
0
,
Subject
to
v
≥
0
,
w
∈
C
′
.



---

*Source: 687917-2014-04-24.xml*