In this paper, a vector optimization problem over cones is considered, where the functions involved are
η
semidifferentiable. Necessary and sufficient optimality conditions are obtained. A dual is formulated and duality results are proved using the concepts of cone
ρ
semilocally preinvex, cone
ρ
semilocally quasipreinvex and cone
ρ
semilocally pseudopreinvex functions.
AMS Mathematics Subject Classification : 15A09, 15A15, 15A18, 65F15, 65F40.
1. Introduction
Ewing
[1]
introduced the concept of semilocally convex functions. It was further extended to semilocally quasiconvex, semilocally pseudoconvex functions by Kaul and Kaur
[2]
. Necessary and sufficient optimality conditions were derived by Kaul and Kaur
[3
,
4]
, and Suneja and Gupta
[8]
.
Weir and Mond
[13]
considered preinvex functions for multiple objective optimization. Further Weir and Jeyakumar
[12]
introduced the class of conepreinvex functions and obtained optimality conditions and duality theorems for a scalar and vector valued programs. Weir
[11]
introduced conesemilocally convex functions and studied optimality and duality theorems for vector optimization problems over cones. Preda and StancuMinasian
[5
,
6
,
7]
studied optimality and duality results for a fractional programming problem where the functions involved were semilocally preinvex.
In the recent years Suneja et al.
[9]
introduced the concepts of
ρ
semilocally preinvex and related functions and obtained optimality and duality for multiobjective nonlinear programming problem, Suneja and Bhatia
[10]
defined conesemilocally preinvex and related functions. They obtained necessary and sufficient optimality conditions for a vector optimization problem over cones. In this paper, we have defined cone
ρ
semilocally preinvex, cone
ρ
semilocally quasipreinvex, cone
ρ
semilocally pseudopreinvex functions and established necessary and sufficient optimality conditions for a vector optimization problem over cones.
2. Definitions and Preliminaries
Let
S
⊆
R^{n}
and
η
:
S
×
S
→
R^{n}
and
θ
:
S
×
S
→
R^{n}
be two vector valued functions.
Definition 2.1
. The set
S
⊆
R^{n}
is said to be
η
locally star shaped set at
x
^{∗}
∈
S
if for each
x
∈
S
there exists a positive number
a _{η}
(
x, x
^{∗}
) ≤ 1 such that
x
^{∗}
+
λ_{η}
(
x,x
^{∗}
) ∈
S
, for 0 ≤
λ
≤
a_{η}
(
x,x
^{∗}
).
Definition 2.2
(
[10]
). Let
S
⊆
R^{n}
be an
η
locally star shaped set at
x
^{∗}
∈
S
and
K
⊆
R^{m}
be a closed convex cone with nonempty interior. A vector valued function
f
:
S
→
R^{m}
is said to be Ksemilocally preinvex (
K
Slpi) at
x
^{∗}
with respect to
η
if corresponding to
x
^{∗}
and each
x
∈
S
, there exist a positive number
d_{η}
(
x,x
^{∗}
) ≤
a_{η}
(
x, x
^{∗}
) ≤ 1 such that
We now introduce
ρ
semilocally preinvex functions over cones.
Definition 2.3.
Let
S
⊆
R^{n}
be an
η
locally star shaped set at
x
^{∗}
∈
S,ρ
∈
R^{m}
and
K
⊆
R^{m}
be a closed convex cone with nonempty interior. A vector valued function
f
:
S
→
R^{m}
is said to be
ρ
semilocally preinvex over K(
kρ
Slpi) at
x
^{∗}
∈
S
with respect to
η
if corresponding to
x
^{∗}
and each
x
∈
S
, there exists a positive number
d_{η}
(
x,x
^{∗}
) ≤
a_{η}
(
x,x
^{∗}
) ≤ 1 such that
Remark 2.1.
If
ρ
= 0 the definition of
Kρ
Slpi function reduces to that of
K
slpi function given by Suneja and Meetu
[10]
.
If
K
=
R
^{+}
, the definition of
Kρ
slpi function reduces to that of
ρ
slpi function given by Suneja et al.
[9]
. In addition if
η
(
x,x
^{∗}
) =
x
−
x
^{∗}
then
Kρ
semilocally preinvex functions reduces to
K
semilocally convex functions defined by Weir
[11]
.
We now give an example of a function which is
Kρ
slpi but fails to be
ρ
slpi.
Example 2.1.
We consider the following
η
locally star shaped set as given by Suneja and Meetu
[10]
. Let
S
=
R
╲
E
, where
θ(
x,x
^{∗}
) =
x
−
x
^{∗}
Consider the function
f
:
S
→
R
^{2}
defined by
Then
f
is
Kρ
slpi at
x
^{∗}
= −1. But
f
is not
ρ
slpi because for
x
= 1,
Definition 2.4.
The function
f
:
S
→
R^{m}
is said to be
η
semidifferentiable at
x
^{∗}
∈
S
if
exists for each
x
∈
S
.
Theorem 2.1.
If f is Kρ

Slpi at x
^{∗}
then
Proof
. Since the function
f
is
Kρ
slpi at
x
^{∗}
with respect to
η
therefore corresponding to each
x
∈
S
there exists a positive number
such that
which implies
Since
K
is a closed cone, therefore by taking limit as
λ
→ 0
^{+}
, we get
We now introduce
Kρ
semilocally naturally quasi preinvex (
Kρ
slnqpi) over cones.
Definition 2.5.
The function
f
is said to be
Kρ
semilocally naturally quasi preinvex (
Kρ
Slnqpi) at
x
^{∗}
with respect to
η
if
Theorem 2.2.
If f is Kρ

slpi at x
^{∗}
∈
S with respect to η then f is Kρ

slnqpi at x
^{∗}
with respect to same η
.
Proof
. Let
f
be
Kρ
slpi at
x
^{∗}
, then there exists a positive number
d _{η}
(
x,x
^{∗}
) ≤
a_{η}
(
x,x
^{∗}
) such that
Suppose that
then
Adding (2.1) and (2.2) we get
Since
K
is a closed cone, therefore taking limit as
λ
→ 0
^{+}
, we get
Thus
But the converse is not true as shown in the following example.
Example 2.2.
Consider set
S
=
R
/
E
, where
E
=
Then as discussed in Example 2.1,
S
is
η
locally star shaped. Consider the function
f
:
S
→
R
^{2}
defined by
θ
(
x, x
^{∗}
) =
x
−
x
^{∗}
.
Then function
f
is
Kρ
slnqpi at
x
^{∗}
= −2, for
ρ
= (1, 0), where
because
But the function
f
fails to be
kρ
slpi at
x
^{∗}
= −2 by Theorem 2.1 because for
x
= 1,
Definition 2.6.
The function
f : S
→
R^{m}
is said to be
Kρ
semilocally quasi preinvex (
Kρ
slqpi) at
x
^{∗}
with respect to
η
if
Remark 2.2.
The following diagram illustrates the relation among
Kρ
slpi function,
Kρ
slnqpi and
Kρ
slqpi functions.
We now give an example of a function which is
Kρ
slnqpi but fails to be
kρ
slqpi.
Example 2.3.
The function
f
considered in Example 2.2 is
Kρ
slnqpi at
x
^{∗}
= −2. But fails to be
Kρ
slqpi at
x
^{∗}
= −2 because for
x
= 1
but
The next definition introduces cone semilocally pseudo preinvex functions over cone.
Definition 2.7.
The function
f : S
→
R^{m}
is said to be
Kρ
semilocally pseudo preinvex (
Kρ
slppi) at
x
^{∗}
, with respect to
η
if
3. Optimality Conditions
Consider the following Vector Optimization Problem

(VOP)Kminimizef(x)

subject to −g(x) ∈Q
where
f
:
S
→
R^{m}
and
g : S
→
R^{p}
are
η
semidifferentiable functions with respect to same
η
and
S
⊆
R^{n}
is a nonempty
η
locally star shaped set.
Let
K
⊆
R^{m}
and
Q
⊆
R^{p}
be closed convex cones having nonempty interior and let
X
= {
x
∈
S
: −
g
(
x
) ∈
Q
} be the set of all feasible solutions of (VOP).
Definition 3.1
. A point
x
^{∗}
∈
X
is called

(i) a weak minimum of (VOP), if for allx∈X, f(x∗) −f(x) ∉ intK.

(ii) a minimum of (VOP), if for allx∈X,f(x∗) −f(x) ∉K\ {0} .

(iii) a strong minimum of (VOP), if for allx∈X,f(x) −f(x∗) ∈K.
We will use the following Alternative Theorem given by Weir and Jeyakumar
[12]
.
Theorem 3.1.
Let X, Y be real normed linear spaces and K be a closed convex cone in Y with nonempty interior, let S ⊆ X. Suppose that f : S → Y be Kpreinvex. Then exactly one of the following holds:

(i)there exists x∈S such that−f(x) ∈ intK,

(ii)there exists0 ≠p∈K∗such that(pTf)(S) ⊆R+,
where
int
denotes interior and K
^{∗}
is the dual cone of K
.
We now establish the necessary optimality conditions for (VOP).
Theorem 3.2
(Fritz John Type Necessary Optimality Conditions).
Let x
^{∗}
∈
X
be a weak minimum of
(VOP)
and suppose
(
df
)
^{+}
(
x
^{∗}
,
η
(
x,x
^{∗}
))
and
(
dg
)
^{+}
(
x
^{∗}
,
η
(
x,x
^{∗}
))
are Kpreinvex and Qpreinvex functions of x respectively with respect to same η
(
x,x
^{∗}
)
and η
(
x
^{∗}
,
x
^{∗}
) = 0
then there exists
τ
*
τ
^{∗}
∈
K
^{∗}
,
µ
^{∗}
∈
Q
^{∗}
such that
Proof
. We assert that the system
has no solution
x
∈
S
, where
F
(
x
) = ((
df
)
^{+}
(
x
^{∗}
,
η
(
x,x
^{∗}
)), (
dg
)
^{+}
(
x
^{∗}
,
η
(
x,x
^{∗}
)) +
g
(
x
^{∗}
)).
If possible, let there be a solution
x
^{0}
∈
S
of (3.3). Then
−
F
(
x
^{0}
) ∈ int(
K
×
Q
) ⇒ −(
df
)
^{+}
(
x
^{∗}
,
η
(
x
^{0}
,
x
^{∗}
)) ∈ int
K
and
−(
dg
)
^{+}
(
x
^{∗}
,
η
(
x
^{0}
,
x
^{∗}
)) −
g
(
x
^{∗}
) ∈ int
Q
.
Since
S
is locally star shaped and
x
^{∗}
,
x
^{0}
∈
S
, therefore we can find
λ
_{0}
> 0 such that for
λ
∈ (0,
λ
_{0}
),
x
^{∗}
+
λη
(
x
^{0}
,
x
^{∗}
) ∈
S
.
By definition of (
df
)
^{+}
(
x
^{∗}
,
η
(
x,x
^{∗}
)) and (
dg
) + (
x
^{∗}
,
η
(
x,x
^{∗}
)), it follows that
−[
f
(
x
^{∗}
+
λη
(
x
^{0}
,
x
^{∗}
)) −
f
(
x
^{∗}
)] ∈ int
K
and
− [
g
(
x
^{∗}
+
λη
(
x
^{0}
,
x
^{∗}
)) −
g
(
x
^{∗}
)] −
g
(
x
^{∗}
) ∈ int
Q
.
⇒
f
(
x
^{∗}
) −
f
(
x
^{∗}
+
λη
(
x
^{0}
,
x
^{∗}
)) ∈ int
K
and
−
g
(
x
^{∗}
+
λη
(
x
^{0}
,
x
^{∗}
)) ∈ int
Q
, for
λ
∈ (0,
λ
_{0}
),
which is a contradiction as
x
^{∗}
is a weak minimum of (VOP). Hence the system (3.3) has no solution
x
∈
S
.
Also
F
is (
K
×
Q
) preinvex on
S
as (
df
)
^{+}
(
x
^{∗}
,
η
(
x,x
^{∗}
)) and (
dg
)
^{+}
(
x
^{∗}
,
η
(
x,x
^{∗}
)) are
K
preinvex and
Q
preinvex on
S
respectively. Therefore, by Theorem 3.1, there exists
τ
^{∗}
∈
K
^{∗}
and
µ
^{∗}
∈
Q
^{∗}
not both zero such that
Taking
x
=
x
^{∗}
, we get
Also
µ
^{∗}
∈
Q
^{∗}
and −
g
(
x
^{∗}
) ∈ Q, implies that
From (3.5) and (3.6), we get
µ^{∗T}g
(
x
^{∗}
) = 0.
From (3.4), we get
τ^{∗T}
(
df
)
^{+}
(
x
^{∗}
,
η
(
x,x
^{∗}
)) +
µ^{∗T}
(
dg
)
^{+}
(
x
^{∗}
,
η
(
x,x
^{∗}
)) ≥ 0, for all
x
∈
S
.
We use the following Slater type constraint qualification to prove the KuhnTucker type necessary optimality conditions for (VOP).
Definition 3.2.
The function
g
is said to satisfy Slater type constraint qualification at
x
^{∗}
if
g
is
Q
preinvex at
x
^{∗}
and there exists
∈
S
such that−
g
(
)∈ int
Q
.
Theorem 3.3
(Kuhn Tucker Type Necessary Optimality Conditions).
Let x
^{∗}
∈
X be a weak minimum of
(VOP)
and suppose
(
df
)
^{+}
(
x
^{∗}
,
η
(
x,x
^{∗}
)) and (
dg
)
^{+}
(
x
^{∗}
,
η
(
x,x
^{∗}
))
are Kpreinvex and Qpreinvex functions of x respectively with respect to the same η
(
x,x
^{∗}
).
Suppose that g is Qslpi at x^{∗} and g satisfies Slater type constraint qualification at x^{∗} and η
(
x
^{∗}
,
x
^{∗}
) = 0,
then there exists
0 ≠
τ
^{∗}
∈
K
^{∗}
,
µ
^{∗}
∈
Q
^{∗}
such that
(3.1)
and
(3.2)
hold
.
Proof
. Since
x
^{∗}
is a weak minimum of (VOP), therefore by Theorem 3.2, there exist
τ
^{∗}
∈
K
^{∗}
,
µ
^{∗}
∈
Q
^{∗}
such that (3.1) and (3.2) hold. If possible, let
τ
^{∗}
= 0, then from (3.1), we get
Since
g
is
Q
slpi at
x
^{∗}
, therefore we have
Adding (3.7) and (3.8) and using (3.2), we get
Again by Slater type constraint qualification, there exists
∈
S
such that
which is a contradiction to (3.9). Hence
τ
^{∗}
≠ 0.
Now we will establish some sufficient conditions for (VOP).
Theorem 3.4.
If x
^{∗}
∈
X, f is Kρslpi and g is Qσslpi at x^{∗} and there exist
0 ≠
τ
^{∗}
∈
K
^{∗}
and µ
^{∗}
∈
Q^{∗} satisfying the conditions
(3.1)
and
(3.2),
then x^{∗} is a weak minimum of
(VOP)
provided
τ^{∗T} ρ
+
µ^{∗T} σ
≥ 0.
Proof
. Suppose that
x
^{∗}
is not a weak minimum of (VOP), then there exists
x
∈
X
such that
f
(
x
^{∗}
) −
f
(
x
) ∈ int
K
.
Since 0 ≠
τ
^{∗}
∈
K
^{∗}
, it follows that
Since
f
is
Kρ
slpi and
g
is
Qσ
slpi at
x
^{∗}
, therefore
and
which contradicts (3.10).
Theorem 3.5.
Let x
∈
X
.
If there exist
0 ≠
τ
^{∗}
∈
K
^{∗}
,
µ
^{∗}
∈
Q
^{∗}
satisfying the conditions
(3.1)
and
(3.2),
g is Qσslqpi at x^{∗} and f is Kρslppi at x^{∗} then x^{∗} is a weak minimum of
(VOP)
provided
τ^{∗T} ρ
+
µ^{∗T} σ
≥ 0 .
Proof
. Let
x
∈
X
and suppose
µ
^{∗}
≠ 0. Then −
g
(
x
) ∈
Q
implies that
µ^{∗T}g
(
x
) ≤ 0.
From condition (3.2), it follows that
µ^{∗T}
(
g
(
x
) −
g
(
x
^{∗}
)) ≤ 0,
which gives that
g
(
x
) −
g
(
x
^{∗}
) ∉ int
Q
.
Also
g
is
Qσ
slqpi at
x
^{∗}
, therefore, we get
If
µ
^{∗}
= 0, then the above inequality holds trivially.
On using (3.1), we have
Since
f
is
Kρ
slppi at
x
^{∗}
, we get
− (
f
(
x
) −
f
(
x
^{∗}
)) ∉ int
K
⇒
f
(
x
^{∗}
) − f (
x
) ∉ int
K
.
Thus
x
^{∗}
is a weak minimum of (VOP).
4. Duality
We associate the following MondWeir type dual with (VOP),
(VOD)
K
maximize
f
(
u
)
Theorem 4.1
(Weak Duality).
Let x
∈
X and
(
u, τ, µ
)
be dual feasible, suppose f is Kρslppi and g is Qσslqpi at u then
f
(
u
) −
f
(
x
) ∉ int
K
,
provided τ ρ
+
µσ
≥ 0.
Proof
. Since
x
∈
X
and (
u, τ, µ
) is dual feasible, therefore, we get
µ^{T}
(
g
(
x
) −
g
(
u
)) ≤ 0
If
µ
≠ 0, then the above inequality gives
g
(
x
) −
g
(
u
) ∉ int
Q
.
Since
g
is
Qσ
slqpi at
u
, we get
If
µ
= 0, then the above inequality holds trivially. Now using (4.1), we get
Since
f
is
Kρ
slppi at
u
, we get
− (
f
(
x
) −
f
(
u
)) ∉ int
K
⇒ (
f
(
u
) −
f
(
x
)) ∉ int
K
.
Thus
u
is a weak minimum of (VOD).
Theorem 4.2
(Strong Duality).
Let x
^{∗}
be a weak minimum of
(VOP), (
df
)
^{+}
(
u,η
(
x,u
))
be Kpreinvex and
(
dg
)
^{+}
(
u,η
(
x,u
))
be Qpreinvex functions on S. Suppose slater type constraint qualification holds at x
^{∗}
.
Then there exist
0 ≠
τ
^{∗}
∈
K^{∗} , µ^{∗}
∈
Q
^{∗}
such that
(
x^{∗} , τ^{∗} , µ^{∗}
)
is feasible for
(VOD).
Moreover, if for each feasible
(
u, τ, µ
)
of
(VOD),
hypothesis of above theorem holds then
(
x^{∗} , τ^{∗} , µ^{∗}
)
is a weak maximum of
(VOD).
Proof
. Since all the conditions of Theorem 3.3 hold, therefore, there exist 0 ≠
τ
^{∗}
∈
K
^{∗}
,
µ
^{∗}
∈
Q
^{∗}
such that (3.1) and (3.2) hold. This implies that (
x^{∗} , τ^{∗} , µ^{∗}
) is feasible for (VOD). If possible let (
x^{∗} , τ^{∗} , µ^{∗}
) be not a weak maximum of (VOD), then there exists (
u, τ, µ
) feasible for (VOD) such that
f
(
u
) −
f
(
x
^{∗}
) ∈ int
K
.
But this is a contradiction to weak duality result as
x
^{∗}
∈
X
and (
u, τ, µ
) is feasible for (VOD). Hence (
x^{∗} , τ^{∗} , µ^{∗}
) must be a weak maximum of (VOD).
Acknowledgements
We express our sincere thanks to Dr. Surjeet K. Suneja, and Dr. Sunila Sharma for their valuable comments and suggestions that helped in improving the quality of our paper.
BIO
Sudha Gupta has completed her Ph.D in 1998 and is working in Laxmibhai College, university of Delhi. Her area of interest include vector optimization and generalized convexity.
Department of Mathematics, Laxmibai College (University of Delhi), Ashok Vihar, Delhi 110052, India.
email: vrindagupta88@gmail.com
Vani Sharma has completed her Ph.D in 2008 and is working in Satyawati College, University of Delhi. Her area of interest include vector optimization and generalized convexity.
Department of Mathematics, Satyawati College (Morning) (University of Delhi), Ashok Vihar, Delhi 110052, India.
email: vani5@rediffmail.com
Mamta Chaudhary is working in Satyawati College, university of Delhi, Delhi. Currently, she is perusing her Ph.D. in Mathematics. Her area of interest include vector optimization and generalized convexity.
Department of Mathematics, Satyawati College (Morning) (University of Delhi), Ashok Vihar, Delhi 110052, India.
email: mam.gupta18@gmail.com
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