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Convexity and its generalizations can be extended to vector case in two
different ways, either by defining the concept component wise or by using
the idea of cones. Cone convexity plays a crucial role in optimization
problems. Cambini [15] introduced several classes of vector valued
functions which are possible extensions of scalar generalized concavity.
These classes are defined using three order relations generated by a cone,
the interior of the cone and the cone without origin. In this chapter we
study cone convex and other related functions in vector optimization
problems. This chapter is divided into four sections.
In section 2.1, the concept of cone semistrictly convex functions is
introduced as a generalization of semistrictly convex functions. Certain
properties of these functions and a relation with cone convex functions
have been established. In section 2.2, sufficient optimality conditions are
proved for a vector optimization problem over topological vector spaces,
involving Gâteaux derivatives by assuming the functions to be cone
subconvex. Further a Mond-Weir type dual is associated with the
optimization problem and weak and strong duality results are proved.
In section 2.3, we introduce cone semilocally preinvex, cone semilocally
quasi preinvex and cone semilocally pseudo preinvex functions and study
their properties. These functions are further used to establish necessary
and sufficient optimality conditions for a vector optimization problem
over cones involving η-semi differentiable functions. The section ends by
Chapter 2
CONE CONVEX AND RELATED FUNCTIONS IN VECTOR OPTIMIZATION
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formulating a Mond-Weir type dual and proving various duality
theorems.
Section 2.4, introduces generalized type-I, generalized quasi type-I,
generalized pseudo type-I, generalized quasi pseudo type-I and
generalized pseudo quasi type-I functions over cones, for a nonsmooth
vector optimization problem, using Clarke’s generalized gradients of
locally Lipschitz functions. Sufficient optimality conditions are
established and a Mond-Weir type dual is associated with the
optimization problem. Weak and strong duality results are proved for the
pair under cone generalized type-I assumptions.
2.1 Cone Semistrictly Convex Functions
Yang [153] defined the class of semistrictly convex functions. These
functions were earlier called explicitly convex by Xue and Sheh [152].
This section begins by introducing the notion of cone semistrictly convex
functions.
Let X be a topological vector space and Y be a locally convex topological
vector space, C ⊂ Y be a pointed closed convex cone with nonempty
interior. The ordering in Y is determined by C. Let Y* be the topological
dual space of Y. The conjugate cone *C of C is defined as
{ }= ∈ ≥ ∀ ∈* * * *: , 0,C y Y y y y C .
The positive conjugate cone *sC of C is given by
{ }= ∈ > ∀ ∈* * * *: , 0, \ {0 }sYC y Y y y y C .
Let ⊆S X be a nonempty open convex set and →:f S Y .
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Definition 2.1.1. The function f is said to be C-semistrictly convex on
S, if for every ∈,x y S , { }− ∉( ) ( ) 0Yf x f y implies
( )+ − − + − ∈( ) (1 ) ( ) (1 ) inttf x t f y f tx t y C , for all ∈ (0,1)t .
Remark 2.1.1. For = nX R , =Y R , +=C R , the above definition reduces
to that of semistrictly convex function [153] (Definition 1.1.8).
Remark 2.1.2. The following examples illustrate that the classes of
cone convex and cone semistrictly convex functions are independent.
Example 2.1.1. Let = − ≤ ≤ ( , ) : ,
2 2x x
C x y y y ,
≠=
=
(0,0), 0,( )
(1,1), 0.x
f xx
Then f is a C-semistrictly convex function. However f is not C-convex
because for = − = =1
1, 1,2
x y t
( )+ − − + − = − − ∉( ) (1 ) ( ) (1 ) ( 1, 1)tf x t f y f tx t y C
Example 2.1.2. Let { }= − ≤ ≤( , ) :C x y x y x , → 2:f R R be defined as
= −2 2( ) ( , )f x x x , then f is C-convex, but f fails to be C-semistrictly convex,
because for = = =3 1
1, ,2 2
x y t ,
− = − = ≠
9 9( ) (1, 1), ( ) , , ( ) ( )
4 4f x f y f x f y and
( ) − + − − + − = ∉
1 1( ) (1 ) ( ) (1 ) , int
16 16tf x t f y f tx t y C .
54
We now discuss some properties of C-convex and C-semistrictly convex
functions in terms of Gâteaux derivatives, defined below
Definition 2.1.2 ([61]). Let x ∈ S, then →:f S Y is said to be Gâteaux
differentiable at x if for any x ∈ X, the limit
→
+ −′ =0
( ) ( )( ) : limx t
f x tx f xf x
t
exists and ′ ⋅( )xf is a continuous linear map from X to Y.
The map ′ ′→: ( )x xf x f x is called Gâteaux derivative of f at x in the
direction x. We observe that
(i) ′ =(0) 0xf
(ii) ′ ′ ′α + β = α + β ∀α β ∈ ∀ ∈( ) ( ) ( ), , and ,x x xf x y f x f y R x y S .
Theorem 2.1.1. Let →:f S Y , be Gâteaux differentiable then f is
C-convex on S if and only if
′− − − ∈( ) ( ) ( )yf x f y f x y C , ∀ x, y ∈ S (2.1)
where ′ −( )yf x y is the Gâteaux derivative of f at y in the direction x − y.
Proof. Firstly, let f be C-convex on S, then for every ∈ ∈, , [0,1]x y S t ,
( )+ − − + − ∈( ) (1 ) ( ) (1 )tf x t f y f tx t y C ,
or, [ ]+ − −
− − ∈( ( )) ( )
( ) ( )f y t x y f y
f x f y Ct
.
Taking limit as → +0t , we get
′− − − ∈( ) ( ) ( )yf x f y f x y C as (f′y)+ (x – y) = f′y (x – y).
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Conversely, let us suppose that (2.1) holds.
Let ∈,x y S , ≤ ≤0 1t , then = + − ∈ˆ (1 )x ty t x S .
By using (2.1) we have
′− − − ∈ˆˆ ˆ( ) ( ) ( )xf x f x f x x C
and ′− − − ∈ˆˆ ˆ( ) ( ) ( )xf y f x f y x C
⇒ ′− − − ∈ˆˆ( ) ( ) ( )xf x f x tf x y C (2.2)
and ( ) ′− − − − ∈ˆˆ( ) ( ) 1 ( ) .xf y f x t f y x C (2.3)
Multiplying (2.2) by −(1 )t and (2.3) by t and adding we get,
( ) ( )− + − − + ∈1 ( ) ( ) (1 )t f x tf y f t x ty C , ∀ ∈ ∈, , [0,1]x y S t .
Hence f is C-convex. o
Corollary 2.1.1. Let f be a Gâteaux differentiable C-convex function,
then
( )( )′ ′− − ∈y xf f y x C , for all ∈,x y S .
Proof. Let ∈,x y S , by Theorem 2.1.1
′− − − ∈( ) ( ) ( )yf x f y f x y C , (2.4)
and ′− − − ∈( ) ( ) ( )xf y f x f y x C . (2.5)
Also, ′ ′α = α( ) ( )x xf x f x , for any real α .
Adding (2.4) and (2.5) and using above relation we get
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′ ′− − − ∈( ) ( )y xf y x f y x C ,
or, ( )( )′ ′− − ∈y xf f y x C . o
In the following theorem we establish a relation between C-semistrictly
convex and C-convex functions.
Theorem 2.1.2. Let →:f S Y be C-semistrictly convex. If there exists
α ∈ (0,1) such that for every ∈,x y S ,
α + − α − α + − α ∈( ) (1 ) ( ) ( (1 ) )f x f y f x y C , (2.6)
then f is C-convex.
Proof. Let us suppose to the contrary that f is not C-convex. Then, there
exist ∈,x y S , ∈ (0,1)t such that
+ − − + − ∉( ) (1 ) ( ) ( (1 ) )tf x t f y f tx t y C .
Thus, there exists ∈* *k C such that
+ − − + − <* , ( ) (1 ) ( ) ( (1 ) ) 0k tf x t f y f tx t y ,
as C is a closed convex cone.
We have two cases:
Case 1: { }− ∉( ) ( ) 0Yf x f y .
Since f is C-semistrictly convex, for ∈,x y S , < <0 1t ,
( )+ − − + − ∈ ⊂( ) (1 ) ( ) (1 ) inttf x t f y f tx t y C C .
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Case 2: { }− ∈( ) ( ) 0Yf x f y .
Put = + −(1 )z tx t y , then using f(x) − f(y) ∈ {0Y},
+ − − + − <* , ( ) (1 ) ( ) ( (1 ) ) 0k tf x t f y f tx t y reduces to
− <
− <
*
*
, ( ) ( ) 0
and , ( ) ( ) 0
k f x f z
k f y f z. (2.7)
(i) Let < < α <0 1t and = + − α α 1 1
t tz x y , then,
= + −(1 )z tx t y
− = α + α α
1t tx y
( ) = α + − + − α α α 1 1
t tx y y
( )= α + − α1 1z y .
Using (2.6) we get
α + − α − ∈1( ) (1 ) ( ) ( )f z f y f z C .
This gives
α + − α − ≥*1, ( ) (1 ) ( ) ( ) 0k f z f y f z . (2.8)
From (2.7) we have
( )− α − >*1 , ( ) ( ) 0k f z f y .
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Adding in (2.8) we get,
− >*1, ( ) ( ) 0.k f z f z (2.9)
Let − α = α −
11
tb
t, < <0 1b
then,
= + − α α 1 1
t tz x y
= + − − α α − − 1
1 1t t z tx
xt t
( )α − α − = + − α α − − α 1 1
t t t tx z x
t t
− α − α
= + − α − α −
(1 ) (1 )1
(1 ) (1 )t t
x zt t
= + −(1 )bx b z .
Since { }− ∉( ) ( ) 0Yf x f z , using C-semistrict convexity of f we get,
+ − − ∈1( ) (1 ) ( ) ( ) intbf x b f z f z C ,
or, + − − >*1, ( ) (1 ) ( ) ( ) 0k bf x b f z f z . (2.10)
From (2.7) we have
− <* , ( ) ( ) 0k bf x bf z .
On adding the above inequality in (2.10) we get
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− <*1, ( ) ( ) 0k f z f z which contradicts (2.9).
(ii) Let < α < <0 1t , that is, − α
< <− α
0 11t
.
Let − α − = + − α − α
21
1 1t t
z x y
then − α + −
=− α2
(1 )1
tx x t yz
or, ( )− α = − α21 z z x
or, = α + − α 2(1 )z x z .
Using (2.6) we get,
α + − α − ∈2( ) (1 ) ( ) ( )f x f z f z C .
That is,
α + − α − ≥*2, ( ) (1 ) ( ) ( ) 0k f x f z f z . (2.11)
Using (2.7) as earlier we get
− >*2, ( ) ( ) 0k f z f z . (2.12)
Let − α
=− α(1 )
ts
t, then < <0 1s , as < α < <0 1t .
Now,
α = − − α − α
2 1 1z
z x
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α − = − − − α − α
11 1
z z ty
t t
( )− αα
= − + − α − α − α
111 (1 ) (1 )
tz y
t t
− α − α
= + − − α − α 1
(1 ) (1 )t t
z yt t
= + −(1 )sz s y .
Again using C-semistrict convexity of f, we have
+ − − ∈2( ) (1 ) ( ) ( ) intsf z s f y f z C ,
or, + − − >*2, ( ) (1 ) ( ) ( ) 0k sf z s f y f z .
Using (2.7), the above inequality reduces to
− <*2, ( ) ( ) 0k f z f z , which contradicts (2.12).
Hence f is C-convex. o
We now consider the vector minimization problem
(UP) C-min ( )f x
∈x S ,
where →:f S Y , and S is a nonempty convex subset of X.
Definition 2.1.3. ∈x S is a global efficient solution of (UP) if, there
exists no ∈x S such that
{ }− ∈( ) ( ) \ 0Yf x f x C .
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Definition 2.1.4. ∈x S is a locally efficient solution of (UP) if there
exists a neighbourhood U of x such that
{ }− ∉( ) ( ) \ 0Yf x f x C for all ∈ ∩x U S .
If f is a C-convex function then, we know that every locally efficient
solution of (UP) is a global efficient solution. This property also holds if f
is C-semistrictly convex function as shown in the following theorem.
Theorem 2.1.3. Let S be a nonempty convex subset of X and let
→:f S Y be C-semistrictly convex on S. If x is a locally efficient solution
of (UP) then x is a global efficient solution of (UP).
Proof. Let x be a locally efficient solution of (UP). Then there exists a
neighborhood U of x such that
{ }− ∉( ) ( ) \ 0Yf x f x C for all ∈ ∩x U S . (2.13)
If x is not a global efficient solution for (UP) then, there exists ∈*x S
such that
{ }− ∈*( ) ( ) \ 0Yf x f x C . (2.14)
⇒ − ∉*( ) ( ) {0 }Yf x f x
Then, by C-semistrict convexity of f, we have, for < <0 1t ,
+ − − + − ∈ ⊂* *( ) (1 ) ( ) ( (1 ) ) inttf x t f x f tx t x C C . (2.15)
From (2.14) we have
{ }− ∈*( ) ( ) \ 0Ytf x tf x C , for < <0 1t . (2.16)
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Adding (2.15) and (2.16), we get, for < <0 1t ,
{ }− + − ∈*( ) ( (1 ) ) \ 0Yf x f tx t x C . (2.17)
For sufficiently small > 0t
+ − = + − ∈ ∩* *(1 ) ( )tx t x x t x x U S .
Thus (2.17) contradicts (2.13). o
2.2 Cone Subconvex Functions: Sufficiency and Duality
Hu and Ling [56] studied the generalized Fritz John and generalized
Kuhn-Tucker necessary conditions in terms of Gâteaux derivatives for a
vector optimization problem in ordered topological vector spaces. This
section is devoted to the study of sufficiency and duality results for this
problem by assuming the functions to be cone subconvex defined as
follows:
Let ⊆S X be a nonempty convex set of a topological vector space X and
⊆C Y be a closed convex pointed cone of a locally convex linear space Y
with nonempty interior.
Definition 2.2.1 ([56]). The function →:f S Y is said to be
C-subconvex on S if there exists ∈ intv C such that for any ∈ (0,1)t and
ε > 0 ,
ε + + − − + − ∈( ) (1 ) ( ) ( (1 ) )v tf x t f y f tx t y C , ∀ ∈,x y S .
Remark 2.2.1. It is evident from the definition that C-convexity ⇒
C-subconvexity. The converse however may not hold true as can be seen
from the following example.
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Example 2.2.1. Let → 2:f S R where S = [−5, 5] be defined as
( )= −2 2( ) ,f x x x , { }= − ≤ ≥( , ) : , 0C x y x y y .
Then f is C-subconvex. However, f fails to be C-convex, as for
= = − =1
1, 1,2
x y t ,
+ − − + − = − ∉( ) (1 ) ( ) ( (1 ) ) (1, 1)tf x t f y f tx t y C
Remark 2.2.2. Clearly C-subconvexity ⇒ C-subconvexlikeness, the
converse however fails.
Example 2.2.2. Let ⊆ →2 2:f A R R where
{ } ( ) ( ){ }= ≥ ≥ + > ∪1 2 1 2 1 2( , ) : 0, 0, 1 1,0 , 0,1A x x x x x x ,
be defined as
( )= +1 2 1 2( , ) 2 , 1f x x x x ; { }= − ≤ ≤ ≥( , ) : , 0C x y x y x y .
Then +( ) intf A C is convex and hence f is C-subconvexlike [62]. But as
A is not a convex set, therefore f is not C-subconvex.
We now discuss optimality conditions for the following problem in terms
of Gâteaux derivatives.
(MP) -minimize ( )C f x
subject to − ∈( )g x D ,
where →:f X Y and →:g X Z ; X is a topological vector space and Y and
Z are locally convex linear spaces. ⊆C Y and ⊆D Z are closed convex
pointed cones with nonempty interiors.
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Let { }= ∈ − ∈2 : ( )X x X g x D be the feasible set of (MP).
Hu and Ling [56] gave the following Kuhn-Tucker type necessary
optimality conditions in terms of Gâteaux derivatives for the problem
(MP) by assuming generalized Slater constraint qualification given
below.
Definition 2.2.2. The constraint map g is said to satisfy generalized
Slater constraint qualification, if there exists ∈ 2x̂ X such that
− ∈ˆ( ) intg x D .
Theorem 2.2.1. Let (f, g) be Gâteaux differentiable at ∈ 2x X and
( )C D× -subconvex on X. If x is a weak minimum solution of (MP) and
g satisfies generalized Slater constraint qualification, then there exists
λ ∈ ** \ {0 },
YC µ ∈ *D such that
′ ′λ + µ =, ( ) , ( ) 0x xf x g x , ∀ ∈x X ,
µ =, ( ) 0g x .
We now give the sufficient optimality conditions for a feasible solution to
be a weak minimum of (MP) in the form of the following theorem.
Theorem 2.2.2. Let (f, g) be Gâteaux differentiable at ∈ 2x X and
( )C D× -subconvex on X. If there exists ** \ {0 }
YCλ ∈ , µ ∈ *D such that
′ ′λ + µ =, ( ) , ( ) 0x xf x g x , ∀ ∈ 2x X , (2.18)
µ =, ( ) 0g x (2.19)
then x is a weak minimum of (MP).
65
Proof. Let if possible, x be not a weak minimum of (MP).
Then, there exists 2x X∈ such that
( ) ( ) intf x f x C− ∈ . (2.20)
Since f is C-subconvex on X, therefore, there exists intu C∈ , such that
for any (0,1)t ∈ , 0ε >
( )( ) (1 ) ( ) (1 )tu tf x t f x f tx t x Cε + + − − + − ∈ ,
that is,
( ( )) ( )
( ) ( )f x t x x f x
u f x f x Ct
+ − − ε + − − ∈ .
Letting 0t +→ , we get
( ) ( ) ( )xu f x f x f x x C′ε + − − − ∈ .
Hence we have
, , ( ) , ( ) , ( ) 0xu f x f x f x x′ε λ + λ − λ − λ − ≥ . (2.21)
Again since g is D-subconvex on X, therefore there exists intv D∈ , such
that for any (0,1), 0t ′∈ ε >
, , ( ) , ( ) , ( ) 0xv g x g x g x x′ ′ε µ + µ − µ − µ − ≥ . (2.22)
Adding (2.21) and (2.22) we get
, , , ( ) ( ) , ( ) , ( )
, ( ) , ( ) 0.x x
u v f x f x g x g x
f x x g x x
′ε λ + ε µ + λ − + µ − µ
′ ′− λ − + µ − ≥ (2.23)
66
From (2.18), (2.19) and the fact that (.)xf ′ and (.)xg′ are linear, we have
from (2.23)
, , , ( ) ( ) , ( ) 0u v f x f x g x′ε λ + ε µ + λ − + µ ≥ . (2.24)
Since, 2x X∈ is a feasible solution of (MP), therefore −g(x) ∈ D which
gives that
, ( ) 0g xµ ≤ .
Now from (2.24) we have
, , , ( ) ( ) , ( ) 0u v f x f x g x′ε λ + ε µ + λ − ≥ − µ ≥ .
Since , ′ε ε are arbitrarily chosen, we get , ( ) ( ) 0f x f xλ − ≤ .
As 0,λ ≠ we have ( ) ( ) intf x f x C− ∉ , which is a contradiction to (2.20).
Hence x is a weak minimum of (MP). o
The following sufficient optimality conditions can be proved on the
similar lines as those of Theorem 2.2.2.
Theorem 2.2.3. Let ( , )f g be Gâteaux differentiable at 2x X∈ and
(C×D)-subconvex on X. If there exist λ ∈ ** \ {0 }
YC , *Dµ ∈ such that
2, ( ) , ( ) 0,
, ( ) 0
x xf x x g x x x X
g x
′ ′λ − + µ − ≥ ∀ ∈
µ =
Then x is a weak minimum of (MP).
In the next theorem we give sufficient optimality conditions for a feasible
solution to be an efficient solution for (MP).
67
Theorem 2.2.4. Let (f, g) be Gâteaux differentiable at 2x X∈ and (C×D)
subconvex on X. If there exist *sCλ ∈ , *Dµ ∈ , ( , ) 0λ µ ≠ such that (2.18)
and (2.19) hold then x is an efficient solution of (MP).
Proof. Let x be not an efficient solution of (MP), then there exists
2x X∈ such that ( ) ( ) \ {0 }Yf x f x C− ∈ .
Since f is C-subconvex and g is D-subconvex on X, therefore, proceeding
on the lines of Theorem 2.2.2, we get
, ( ) ( ) 0f x f xλ − ≤ ,
that is { }( ) ( ) \ 0Yf x f x C− ∉ , which is a contradiction. Hence x is an
efficient solution of (MP). o
We associate the following Mond-Weir type dual with the problem (MP)
and study duality results.
(MD) -maximize ( )C f u
subject to ′ ′λ − + µ − ≥ ∀ ∈ 2, ( ) , ( ) 0,u uf x u g x u x X
, ( ) 0g uµ ≥ ,
* *, 0 ,D C u Xµ ∈ ≠ λ ∈ ∈ .
Theorem 2.2.5 (Weak Duality Theorem). Let x be a feasible solution
for (MP) and ( , , )u λ µ be feasible for (MD), (f, g) be Gâteaux differentiable
at 2x X∈ and (C×D)-subconvex on X then,
( ) ( ) intf u f x C− ∉ .
68
Proof. Since g is D-subconvex on X, therefore, there exists intv D∈ such
that for ,x u X∈ , (0,1)t ∈ and 0′ε > ,
( )( ) (1 ) ( ) (1 )tv tg x t g u g tx t u D′ε + + − − + − ∈ ,
or, ( ( )) ( )
( ) ( )g u t x u g u
v g x g u Dt
+ − − ′ε + − − ∈ .
Let 0t +→ , as D is a closed cone, we get
( )( ) ( ) uv g x g u g x u D′ ′ε + − − − ∈ ,
which implies
, , ( ) , ( ) , ( ) 0uv g x g u g x u′ ′ε µ + µ − µ − µ − ≥ . (2.25)
Similarly as f is C-subconvex on X, there exists intw C∈ , such that for
0ε > ,
, , ( ) , ( ) , ( ) 0uw f x f u f x u′ε λ + λ − λ − λ − ≥ . (2.26)
Adding (2.25) and (2.26) and using feasibility of x and ( ), ,u λ µ for (MP)
and (MD) respectively, we get
, , , ( ) ( ) 0w v f x f u′ε λ + ε µ + λ − ≥ .
Since , ′ε ε are arbitrary, we get , ( ) ( ) 0f u f xλ − ≤ .
As *0 C≠ λ ∈ , we have ( ) ( ) intf u f x C− ∉ . o
Theorem 2.2.6 (Strong Duality Theorem). Let (f, g) be Gâteaux
differentiable and (C × D) -subconvex on X. Suppose that g satisfies the
generalized Slater constraint qualification. If x is a weak minimum
69
solution of (MP), then there exist * *0 ,C D≠ λ ∈ µ ∈ such that ( ), ,x λ µ is a
weak maximum solution of (MD).
Proof. Since x is a weak minimum of (MP), therefore by Theorem 2.2.1,
there exist *0 C≠ λ ∈ , *Dµ ∈ such that
, ( ) , ( ) 0x xf x g x x X′ ′λ + µ = ∀ ∈ ,
, ( ) 0g xµ = .
In particular, for x X∈
′ ′λ + µ =, ( ) , ( ) 0x xf x g x
⇒ , ( ) ( ) , ( ) ( ) 0x x x xf x f x g x g x′ ′ ′ ′λ − + µ − =
⇒ , ( ) , ( ) 0x xf x x g x x′ ′λ − + µ − =
Thus, ( ), ,x λ µ is feasible for (MD).
Let if possible x be not a weak maximum of (MD) then, there exists a
feasible solution ( ), ,u λ µ of (MD) such that
( ) ( ) intf u f x C− ∈ ,
which contradicts the Weak Duality Theorem (2.2.5) as x is feasible for
(MP) and ( ), ,u λ µ for (MD).
Thus x must be a weak maximum of (MD). o
70
2.3 Vector Optimization with Cone Semilocally
Preinvex Functions
Preda [114] studied optimality and duality for a multiobjective fractional
programming problem involving semilocally preinvex functions. Later
Suneja et al. [138] introduced ρ-semilocally preinvex, ρ-semilocally quasi
preinvex and ρ-semilocally pseudo preinvex functions and proved
optimality conditions and duality results for a multiobjective non linear
programming problem using the above defined functions. This section
begins by introducing cone semilocally preinvex functions as follows:
Let S nR⊆ be an η-locally star shaped set at , mx S K R∈ ⊆ be a closed
convex cone with nonempty interior and let : mf S R→ be a vector valued
function.
Definition 2.3.1. The function f is said to be K-semilocally preinvex
(K-slpi) at x ∈ S with respect to η if corresponding to each x S∈ , there
exists a positive number ( , )d x xη ≤ ( , )a x xη such that
( ) (1 ) ( ) ( ( , )) , for 0 ( , ).tf x t f x f x t x x K t d x xη+ − − + η ∈ < <
f is said to be K-slpi on S if it is K-slpi at each x S∈ .
The following theorem gives a characterization of cone semilocally
preinvex functions.
Theorem 2.3.1. The function f is K-semilocally preinvex with respect to
η if and only if its epigraph Epi( f ) = {(x, y): x ∈ S, y ∈ f(x)+K} ⊂ n mR R×
is η-locally star shaped in the first component and locally star shaped in
the second component.
71
Proof. First suppose that f is K-slpi on S, with respect to η.
Let 1 1( , )x y and 2 2( , )x y ∈ Epi( ),f then
1 2 1 1, , ( )x x S y f x K∈ ∈ + and 2 2( ) ,y f x K∈ +
which implies that 1 1 1( )y f x k= + and y2 = f (x2) + k2 where k1, k2 ∈ K.
Since f is K-slpi on S, there exists a positive number 1 2( , ) 1a x xη ≤ such
that
x2 + tη 1 2( , )x x ∈ S for 0 < t < aη 1 2( , )x x and there exists a positive number
1 2 1 2( , ) ( , )d x x a x xη η≤ such that
1 2 2 1 2 1 2( ) (1 ) ( ) ( ( , )) , for 0 ( , )tf x t f x f x t x x K t d x xη+ − − + η ∈ < <
1 1 2 2 2 1 2( ) (1 )( ) ( ( , ))t y k t y k f x t x x K⇒ − + − − − + η ∈
1 2 2 1 2 1 2(1 ) ( ( , )) ( (1 ) )ty t y f x t x x K tk t k K⇒ + − − + η ∈ + + − =
1 2 2 1 2(1 ) ( ( , ))ty t y f x t x x K⇒ + − ∈ + η +
( )2 1 2 1 2( , ), (1 ) Epi( )x t x x ty t y f⇒ + η + − ∈ , for 0 < t < dη 1 2( , )x x
⇒ Epi( f ) is η-locally star shaped in first component and locally star
shaped in second component.
Conversely, let Epi( f ) be η-locally star shaped in the first component
and locally star shaped in the second component. Let 1 2,x x ∈ S, then
1 1 2 2( , ( )),( , ( )) Epi ( ).x f x x f x f∈
Thus there exists a positive number 1 2( , ) 1a x xη ≤ such that
72
2 1 2 1 2 1 2( ( , ), ( ) (1 ) ( )) Epi ( ), for 0 ( , )x t x x tf x t f x f t a x xη+ η + − ∈ < <
1 2 2 1 2( ) (1 ) ( ) ( ( , ))tf x t f x f x t x x K⇒ + − ∈ + η +
1 2 2 1 2( ) (1 ) ( ) ( ( , )) ,tf x t f x f x t x x K⇒ + − − + η ∈ for 0 < t < aη 1 2( , )x x .
Hence f is K-slpi on S. o
Remark 2.3.1. If m = n and nK R+= then K-semilocally preinvex
functions reduce to semilocally preinvex functions defined by Preda
[114]. We now give an example of a function which is K-slpi but fails to
be slpi.
Example 2.3.1. Consider the set 1 1
\ where = , {2}2 2
S R E E− = ∪
.
Thus the set S is η-locally starshaped, where
1 1, , , 2, 2 or ,
2 2( , )1 1 1 1
, , 2, or , 22 2 2 2
x x x x x x x xx x
x x x x x x x x
− > ≠ ≠ < −η = − > ≠ < − > ≠ < −
and
2 1 1 1, if 2, 2 or 2,
2 2 2( , ) 2 1
, if 2 , 22
1, elsewhere
xx x x x
x xa x x x
x xx x
η
−< < < < < < − −= −
< < < −
Consider the function :f S R→ 2 defined by
1( ,0), , 2
2( )1
(0, ),2
x x xf x
x x
< ≠= − < −
73
Then, f is K-slpi at 1x = − , where K = {(x, y): y ≤ 0, y ≤ − x}; because
( ) (1 ) ( ) ( ( , ))tf x t f x f x t x x+ − − + η ∈ K, for 0 < t < dη ( , ) ( , )x x a x xη= .
The function f fails to be slpi at 1x = − because for x = 1, there does not
exist any positive number ( , ) ( , )d x x a x xη η≤ such that
( ) (1 ) ( ) ( ( , ))tf x t f x f x t x x+ − − + η 0≥ for 0 ( , )t d x xη< < . o
Remark 2.3.2. If η(x, )x x x= − then K-semilocally preinvex functions
reduce to K-semilocally convex functions defined by Weir [145].
We now give an example of a K-slpi function which fails to be
K-semilocally convex.
Example 2.3.2. The function f considered in Example 2.3.1 is K-slpi at
1x = − but it fails to be K-semilocally convex at 1x = − because for x = 1,
there does not exist any positive real number < 1d such that
( ) (1 ) ( ) ( (1 ) )tf x t f x f tx t x+ − − + − ∈ K for < <0 t d
Theorem 2.3.2. Let : mf S R→ be K-slpi on an η-locally star shaped set
S nR⊆ . Then the set ( )Z f S K= + is locally star shaped.
Proof. Let , ,z z Z∈ then there exist , , ,x x S k k K∈ ∈ such that
= + = +( ) , ( )z f x k z f x k . (2.27)
Since S is an η-locally star shaped set and ,x x ∈ S, therefore there
exists a maximum positive number aη( ,x x ) ≤ 1 such that
( , )x t x x S+ η ∈ for 0 < t < aη( ,x x ).
74
As f is K-slpi on S, there exists a positive number ( , ) ( , )d x x a x xη η≤ such
that
( ) (1 ) ( ) ( ( , )) , for 0 ( , )tf x t f x f x t x x K t d x xη+ − − + η ∈ < <
( ) (1 ) ( ) ( ( , ))tf x t f x f x t x x K⇒ + − ∈ + η + .
On using (2.27) we have
( ) (1 )( ) ( )t z k t z k f S K− + − − ∈ +
(1 ) ( ) (1 )tz t z f S tk t k K⇒ + − ∈ + + − +
(1 ) ( )tz t z f S K⇒ + − ∈ + for 0 < t < dη( ,x x ) ,
as K is a convex cone.
Hence Z is locally star shaped. o
Theorem 2.3.3. Let f be K-slpi on S with respect to η then for each
,my R∈ the set { }( ) : ( )fS y x S y f x K= ∈ ∈ + is η-locally star shaped.
Proof. Let x, ( ), ,mfx S y y R∈ ∈ then x, x S∈ such that
( )y f x K∈ + and ( ) ,y f x K∈ +
Thus there exist ,k k K∈ such that
( ) and ( )y f x k y f x k= + = + . (2.28)
Since f is K-slpi with respect to η therefore there exists a positive number
( , )d x xη ≤ ( , )a x xη where : nS S R× →η such that
( ) (1 ) ( ) ( ( , )) , 0 ( , )tf x t f x f x t x x K t d x xη+ − − + η ∈ < <
75
On using (2.28) we get
( ) (1 )( ) ( ( , )) , 0 ( , )t y k t y k f x t x x K t d x xη− + − − − + η ∈ < <
( (1 ) ) ( ( , ))y tk t k f x t x x K⇒ − + − − + η ∈
( ( , )) , for 0 ( , )y f x t x x K t d x xη⇒ − + η ∈ < <
( , ) ( ), for 0 ( , )fx t x x S y t d x xη⇒ + η ∈ < <
Hence Sf (y) is η-locally star shaped. o
We now give the definition of η-semi differentiable function.
Definition 2.3.2. The function f : S mR→ is said to be η-semi
differentiable at x ∈ S if
[ ]0
1( ) ( , ( , )) lim ( ( , )) ( )
tdf x x x f x t x x f x
t+
+
→η = + η − exists for each x ∈ S.
Remark 2.3.3.
(1) If ( , )x x x x= −η then η-semi differentiability of f reduces to (one
sided) directional differentiability of f at x in the direction x − x , as
considered by Weir [145].
(2) If m = 1 and ( , )x x x x= −η , then η-semi differentiability reduces to
semi differentiability [72].
Let f : S → Rm be η-semi differentiable at x S∈ .
In the following result, we give another property of K-slpi functions.
Theorem 2.3.4. If f is K-slpi at x then ( ) ( ) ( ) ( , ( , ))f x f x df x x x K+− − η ∈
x S∀ ∈ .
76
Proof. Since the function f is K-slpi at x with respect to η, therefore
corresponding to each x S∈ there exists a positive number
( , ) ( , )d x x a x xη η≤ such that
( ) (1 ) ( ) ( ( , )) , for 0 ( , )tf x t f x f x t x x K t d x xη+ − − + η ∈ < <
which implies
[ ]1( ) ( ) ( ( , )) ( ) , 0 ( , )f x f x f x t x x f x K t d x x
t η− − + η − ∈ < <
Since K is a closed cone, therefore taking limit as 0t +→ , we get
( ) ( ) ( ) ( , ( , ))f x f x df x x x K+− − η ∈ , x S∀ ∈ . ð
We now introduce semilocally naturally quasi preinvex functions over
cones.
Definition 2.3.3. The function f is said to be K-semilocally naturally
quasi preinvex (K-slnqpi) at ,x with respect to η if
( ( ) ( )) ( ) ( , ( , )) .f x f x K df x x x K+− − ∈ ⇒ − η ∈
Remark 2.3.4. If m = n and cone K = ,nR+ K-slnqpi functions reduce to
slqpi functions defined by Preda [114].
Theorem 2.3.5. If the set { }( ) : ( )fS y x S y f x K= ∈ ∈ + is η-locally star
shaped for each y ∈ mR , then f is K-semilocally naturally quasi preinvex
on S with respect to same η.
Proof. Let Sf (y) be η-locally star shaped for each y ∈ mR .
Let x, x ∈ S such that − ( f(x) – f( x )) ∈ K.
Denoting ( ),y f x= we get − ( f(x) −y) ∈ K.
77
( ) ,y f x K⇒ ∈ +
( )fx S y⇒ ∈
Also ( )fx S y∈ as 0 ∈ K.
Since Sf (y) is η-locally star shaped, therefore there exists a maximum
positive number ( , )a x xη ≤ 1 such that
( , ) ( ),fx t x x S y+ η ∈ for 0 < t < aη (x, x )
which implies,
( ( , ))y f x t x x K∈ + η + for 0 < t < aη (x, x ).
Thus,
( ) ( ( , )) , for 0 ( , )f x f x t x x K t a x xη∈ + η + < <
( ( ( , )) ( )) ,f x t x x f x K⇒ − + η − ∈ for 0 ( , ).t d x xη< <
1( ( ( , )) ( )) , 0 ( , )f x t x x f x K t d x x
t η−
⇒ + η − ∈ < <
Since K is a closed cone, therefore taking limit as 0t +→ , we get
( ) ( , ( , ))df x x x K+− η ∈
Thus ( ( ) ( ))f x f x K− − ∈ ( ) ( , ( , ))df x x x K+⇒ − η ∈ .
Hence f is K-slnqpi on S. ð
Theorem 2.3.6. If f is K-slpi at x S∈ with respect to η then f is
K-slnqpi at ,x with respect to same η.
78
Proof. Let f be K-slpi at x , then there exists a positive number
( , ) ( , )d x x a x xη η≤ such that
( ) (1 ) ( ) ( ( , )) ,tf x t f x f x t x x K+ − − + η ∈ for 0 ( , )t d x xη< < . (2.29)
Suppose that,
( ( ) ( ))f x f x K− − ∈
then,
( ( ) ( )) , for 0t f x f x K t− − ∈ > . (2.30)
Adding (2.29) and (2.30) we have
[ ]( ( , )) ( )) , for 0 ( , )f x t x x f x K t d x xη− + η − ∈ < < .
1( ( ( , )) ( )) , 0 ( , )f x t x x f x K t d x x
t η−
⇒ + η − ∈ < <
Since K is a closed cone, therefore taking limit as 0t +→ , we get
( ) ( , ( , ))df x x x K+− η ∈
Thus
( ( ) ( ))f x f x K− − ∈ ( ) ( , ( , ))df x x x K+⇒ − η ∈ .
Hence f is K-slnqpi at x with respect to same η. ð
The converse of the above theorem may not hold as can be seen from the
following example.
Example 2.3.3. Consider the set = \S R E , where 1 1
, {2}2 2
E = − ∪ .
Then as discussed in Example 2.3.1, S is η-locally starshaped.
79
Consider the function 2:f S R→ defined by
2 1( ,0),
2( )1
(0, ), , 2.2
x xf x
x x x
− < −= − > ≠
The function f is K-slnqpi at 2x = − , where {( , ) : 0, }K x y y y x= ≤ ≥ ,
because
( ( ) ( ))f x f x K− − ∈
⇒ 1
22
x− ≤ < −
⇒ ( ) ( , ( , )) ( 4( 2),0)df x x x x K+− η = − + ∈
The function f fails to be K-slpi at 2x = − by Theorem 2.3.4, because for
1x =
( ) ( ) ( ) ( , ( , )) (16, 1)f x f x df x x x K+− − η = − ∉
Definition 2.3.4. The function : mf S R→ is said to be K-semilocally
quasi preinvex (K-slqpi) at x with respect to η, if
( ( ) ( )) int ( ) ( , ( , )) .f x f x K df x x x K+− ∉ ⇒ − η ∈
Theorem 2.3.7. If K is a pointed cone and f is K-slqpi at x then f is
K-slnqpi at x with respect to same η.
Proof. Let K be a pointed cone and f be K-slqpi at x with respect to η,
then,
( ( ) ( )) int ( ) ( , ( , ))f x f x K df x x x K+− ∉ ⇒ − η ∈ . (2.31)
80
Suppose that
( ( ) ( ))f x f x K− − ∈ . (2.32)
Since K is pointed, ( ) {0}K K∩ − =
⇒ int K ( )K∩ − = φ
⇒ − K \ intmR K⊂ .
In view of (2.32) we get ( ) ( ) int .f x f x K− ∉
Thus by (2.31) we have ( ) ( , ( , ))df x x x K+− η ∈ .
Hence f is K-slnqpi. ð
The converse of the above theorem may not hold, as can be seen by the
following example.
Example 2.3.4. The function f considered in Example 2.3.3 is K-slnqpi
at 2x = − . But f fails to be K-slqpi at 2x = − because for 1x =
( ) ( ) (4, 1) intf x f x K− = − ∉
where as ( ) ( , ( , )) (12,0)df x x x K+− η = ∉ .
The next definition introduces cone semilocally pseudo preinvex
functions.
Definition 2.3.5. The function : mf S R→ is said to be K-semilocally
pseudo preinvex (K-slppi) at ,x with respect to η if
( ) ( , ( , )) intdf x x x K+− η ∉ ⇒ ( ( ) ( ))f x f x− − ∉ int K
We consider the vector optimization problem
81
(VOP) K-minimize f (x)
subject to ( )g x Q− ∈
h(x) = {0 }kR
where f : , : and :m p kS R g S R h S R→ → → are η-semi differentiable
functions with respect to same η and nS R⊆ is a nonempty η-locally star
shaped set.
Let mK R⊆ and pQ R⊆ be closed convex cones having nonempty
interior and let { }3 : ( ) , ( ) {0 }kRX x S g x Q h x= ∈ − ∈ = be the set of all
feasible solutions of (VOP).
Let 0 ( , , ) :F f g h S S′= → where nS R⊆ is a nonempty set and
m p kS R R R′ = × × and 0 ( {0 })kRK K Q= × × . If 0F is 0K -slpi on S , that
is, f is K-slpi, g is Q-slpi and h is {0 }kR-slpi with respect to same
η , then by Theorem 2.3.2, 0 0( )F S K+ is locally starshaped. If we
assume 0 0( )F S K+ to be a closed set then 0 0( )F S K+ becomes convex
and the following alternative theorem follows on the lines of Illés
and Kassay [60].
Theorem 2.3.8 (Theorem of Alternative). Let 0F be 0K -slpi on S
such that 0 0( )F S K+ is closed with nonempty interior, then the following
assertions hold:
(i) if there is no x S∈ such that ( ) intf x K∈ − , ( )g x Q∈ − and
( ) {0 }kRh x = , then there exist K +∈λ , Q+∈µ and kR∈ν with
( , , ) 0≠λ µ ν such that
( ) ( ) ( ) 0T T Tf x g x h x+ + ≥λ µ ν , ∀ x S∈ .
82
(ii) if there exist \ {0}K +∈λ , Q+∈µ and kR∈ν such that
( ) ( ) ( ) 0T T Tf x g x h x+ + ≥λ µ ν , ∀ x S∈ ,
then there is no x S∈ such that
( ) intf x K∈ − , ( )g x Q∈ − and ( ) {0 }kRh x = .
We shall be using the following constraint qualification to prove the
necessary optimality conditions for (VOP).
Definition 2.3.6. The constraint pair (g, h) is said to satisfy
generalized Slater type constraint qualification at x if (g, h) is
( {0 })-slpi atkRQ x× and there exists *x S∈ such that
*( ) intg x Q− ∈ and *( ) {0 }kRh x = .
We now establish the necessary optimality conditions for (VOP).
Theorem 2.3.9 (Necessary Optimality Conditions).
Let = −1( ) ( ( ) ( ), ( ), ( ))F x f x f x g x h x x S∀ ∈ and 1( ) ( {0 })kRF S K Q+ × × be
closed with nonempty interior. Let ∈ 3x X be a weak minimum of (VOP),
f be K-slpi, g be Q-slpi and h be {0 }kR-slpi with respect to same η.
Suppose that the pair (g, h) satisfies generalized Slater type constraint
qualification and ( , ) 0,x x =η then there exist 0 , K Q+ +≠ λ ∈ µ ∈ and
kRν ∈ such that
( ) ( , ( , )) ( ) ( , ( , ))T Tdf x x x dg x x x+ +λ η + µ η + ( ) ( , ( , )) 0T dh x x x+ν η ≥ ,
for all x S∈ . (2.33)
and µ =( ) 0T g x . (2.34)
83
Proof. Since x is a weak minimum of (VOP), therefore there does not
exist any x S∈ such that
( ) ( ) intf x f x K− ∈ −
( )g x Q∈ −
and ( ) {0 }kRh x = .
Then by Theorem 2.3.8, there exist ,K +λ ∈ , kQ R+µ ∈ ν ∈ , ( , , ) 0λ µ ν ≠
such that
( ( ) ( )) ( ) ( ) 0, for allT T Tf x f x g x h x x Sλ − + µ + ν ≥ ∈
( ) ( ) ( ) ( ) forallT T T Tf x g x h x f x x S⇒ λ + µ + ν ≥ λ ∈ (2.35)
Now, and ( ) ,Q g x Q+µ ∈ − ∈ therefore ( ) 0T g xµ ≤ .
By taking x = x in (2.35) and using ( ) 0h x = , we get ( ) 0T g xµ ≥ .
Thus
( ) 0T g xµ = . (2.36)
From (2.35), (2.36) and ( ) 0h x = , we have
( )( ) ( )( ) 0T T T T T Tf g h x f g h xλ + µ + ν − λ + µ + ν ≥ , for all x S∈
As ( , ) ,x t x x S+ η ∈ for 0 < t < aη( , )x x we have
( )( )( , ) ( )( ) 0T T T T T Tf g v h x t x x f g h xλ + µ + + η − λ + µ + ν ≥
which can be rewritten as,
( ( ( , )) ( )) ( ( ( , )) ( ))T Tf x t x x f x g x t x x g xλ + η − + µ + η −
84
( ( ( , )) ( )) 0T h x t x x h x+ ν + η − ≥
Dividing by t > 0 and taking limit as t → 0+ we get
+ + +λ η + µ η + ν η ≥( ) ( , ( , )) ( ) ( , ( , )) ( ) ( , ( , )) 0,T T Tdf x x x dg x x x dh x x x
for all x ∈ S. (2.37)
Next, let if possible 0λ = , then (2.37) reduces to,
( ) ( , ( , )) ( ) ( , ( , )) 0T Tdg x x x dh x x x+ +µ η + ν η ≥ for all x ∈ S. (2.38)
Since (g, h) is ( { })kRQ × 0 -slpi at x , therefore we have for every x ∈ S,
( ) ( ) ( ) ( , ( , ))g x g x dg x x x Q+− − η ∈
and ( ) ( ) ( ) ( , ( , )) {0 }kRh x h x dh x x x+− − η =
( ) ( ) ( ) ( , ( , )) 0T T Tg x g x dg x x x+⇒ µ − µ − µ η ≥ (2.39)
and ( ) ( ) ( ) ( , ( , )) 0T T Th x h x dh x x x+ν − ν − ν η = . (2.40)
Adding (2.39), (240) and using (2.36) and h( ) 0kx = , we get
( ) ( ) ( ) ( , ( , ))T T Tg x h x dg x x x+µ + ν − µ η ( ) ( , ( , )) 0,T dh x x x+−ν η ≥
for all x ∈ S. (2.41)
On using (2.38) we obtain,
( ) ( ) 0,T Tg x h xµ + ν ≥ for all x ∈ S. (2.42)
Now, by generalized Slater type constraint qualification, there exists *x S∈ such that,
85
*( )g x− ∈ int Q and h(x*) {0 }kR=
which implies *( ) 0T g xµ < and *( ) 0Th xν = .
Adding the above we have
* *( ) ( ) 0T Tg x h xµ + ν < (2.43)
which is a contradiction to (2.42).
Hence 0λ ≠ . o
The following theorem establishes sufficiency result for (VOP).
Theorem 2.3.10 (Sufficient Optimality Conditions). Let 3x X∈ and f
be K-slppi, g be Q-slqpi and h be {0 }kR-slnqpi at x with respect to same
η . If there exist 0 , and such thatkK Q R+ +≠ λ ∈ µ ∈ ν ∈ (2.33) holds
∀ x ∈ X3 and (2.34) is satisfied, then x is a weak minimum of (VOP).
Proof. Let x be feasible for (VOP) then ( )g x− ∈ Q.
On using ,Q+µ ∈ we get ( ) 0.T g xµ ≤ (2.44)
In view of (2.34), (2.44) can be written as ( ( ) ( )) 0T g x g xµ − ≤ (2.45)
If µ ≠ 0, then from (2.45), ( ) ( ) intg x g x Q− ∉
Since g is Q-slqpi at x , we get
( ) ( , ( , ))dg x x x Q+− η ∈
which gives,
( ) ( , ( , )) 0T dg x x x+µ η ≤ . (2.46)
If 0µ = then (2.46) holds trivially.
86
Again for 3 , ( ) {0 }kRx X h x∈ = , therefore
( ( ) ( )) {0 }kRh x h x− − =
Since h is {0 }kR-slnqpi at x we have ( ) ( , ( , )) {0 }kR
dh x x x+− η = .
Therefore
( ) ( , ( , )) 0T dh x x x+ν =η (2.47)
Adding (2.46) and (2.47) and using (2.33) we get ( ) ( , ( , )) 0T df x x x+λ ≥η .
Since 0λ ≠ , we obtain
( ) ( , ( , ))df x x x+− η ∉ int K.
As f is K-slppi we get
− − ∉( ( ) ( ))f x f x int K, that is − ∉( ) ( ) intf x f x K .
Since x ∈ X3 is arbitrarily chosen, therefore x is a weak minimum
of (VOP). o
The following Mond-Weir type dual is associated with the primal
problem (VOP).
(VOD) K-maximize f (u)
subject to
( ) ( , ( , )) ( ) ( , ( , ))T Tdf u x u dg u x u+ +λ η + µ η + ( ) ( , ( , )) 0T dh u x u+ν η ≥
∀ x ∈ X3 (2.48)
( ) 0T g uµ ≥ (2.49)
( ) 0 kRh u = (2.50)
87
0 ,K Q+ +≠ λ ∈ µ ∈ and ν ∈ Rk , u ∈ S
Theorem 2.3.11 (Weak Duality). Let x be feasible for (VOP) and
( , , , )u λ µ ν be feasible for (VOD). Let f be K-slppi, g be Q-slqpi and h be
{0 }kR-slnqpi at u, with respect to same η. Then
f (u) − f (x) ∉ int K.
Proof. Since x is feasible for (VOP) and ( , , , ) feasible for (VOD)u λ µ ν
therefore we have,
( ) 0T g xµ ≤ and ( ) 0T g uµ ≥
which implies
( ( ) ( )) 0T g x g uµ − ≤ .
If 0µ ≠ , then the above inequality results in g( x ) − g(u) ∉ int Q.
Since g is Q-slqpi we have,
( ) ( , ( , ))dg u x u Q+− ∈η
That is,
( ) ( , ( , )) 0T dg u x u+µ ≤η . (2.51)
(2.51) also holds if 0µ = .
Again using feasibility of x and u, we have ( ( ) ( )) {0 }kRh x h u− − = .
As h is {0 }kR-slnqpi at u, we get
( ) ( , ( , )) {0 }kRdh u x u+− η =
88
therefore, ( ) ( , ( , )) 0T dh u x u+ν =η . (2.52)
Using (2.51), (2.52) in (2.48) we obtain
( ) ( , ( , )) 0T df u x u+λ ≥η .
As, 0 K +≠ λ ∈ we have
( ) ( , ( , ))df u x u+− ∉η int K,
Because f is K-slppi at u, we get − (f( x ) − f(u)) ∉ int K which gives
(f(u) − f( x )) ∉ int K. ð
Theorem 2.3.12 (Strong Duality). Let f be K-slpi, g be Q-slpi and h
be {0 }kR-slpi with respect to same η. Let F1(S) + (K × Q × {0 }kR
) be
closed with nonempty interior. Suppose that the pair (g, h) satisfies
generalized Slater type constraint qualification. If x is a weak
minimum of (VOP) and ( , ) 0x x =η , then there exist *0 K +≠ λ ∈ , * Q+µ ∈ and * kRν ∈ such that * * *( , , , )x λ µ ν is a feasible solution of
(VOD). Moreover if the conditions of Weak Duality Theorem 2.3.11
are satisfied for all feasible solutions of (VOP) and (VOD) then * * *( , , , )x λ µ ν is a weak maximum of (VOD).
Proof. Since x is a weak minimum of (VOP), therefore by
Theorem 2.3.9, there exist * * *0 , , kK Q R+ +≠ λ ∈ µ ∈ ν ∈ such that
* *( ) ( , ( , )) ( ) ( , ( , ))T Tdf x x x dg x x x+ +λ η + µ η * ( ) ( , ( , )) 0T dh x x x++ ν η ≥ , ∀ ∈x S
and
* ( ) 0T g xµ = .
Thus * * *( , , , )x λ µ ν is feasible for (VOD). Let if possible, * * *( , , , )x λ µ ν be
not a weak maximum of (VOD). Then there exists ( , , , )u λ µ ν feasible for
89
(VOD) such that f (u) − f ( x )∈ int K which contradicts Weak Duality
Theorem (Theorem 2.3.11) as x is feasible for (VOP). ð
2.4 Optimality and Duality in Vector Optimization
Involving Generalized Type-I Functions Over Cones
Based on the work of Craven [25], Reiland [117] extended the concept
of invexity to nonsmooth Lipschitz functions. Yen and Sach [159]
defined cone invex and cone invex in the limit. Suneja et al. [137] called
these functions cone generalized invex and cone nonsmooth invex. They
further introduced the concepts of cone nonsmooth quasi invex, cone
nonsmooth pseudo invex and other related functions in terms of Clarke’s
[22] generalized directional derivatives and used them to obtain
optimality and duality results for a nonsmooth vector optimization
problem.
In this section we introduce generalized type-I and nonsmooth type-I
functions over cones and study optimality and duality results for the
problem:
(VP) K-minimize f (x)
subject to − g(x) ∈ Q
where f : S → mR , g : S → pR , are locally Lipschitz functions on S = nR ,
K and Q are closed convex pointed cones with nonempty interiors in mR
and pR respectively. Let X0 = { x∈ nR : −g(x) ∈ Q} be the feasible set of
(VP).
We now give the definitions of cone generalized type-I and cone non
smooth type-I functions.
Definition 2.4.1. (f, g) is said to be (K×Q) generalized type-I at the
point u ∈ nR , if there exists η : X0 × nR → nR such that for every x ∈ X0
and A ∈ ∂f(u), B ∈ ∂g(u),
90
f(x) − f(u) − Aη(x, u) ∈ K
− g(u) − Bη(x, u) ∈ Q.
Definition 2.4.2. (f, g) is said to be (K×Q) nonsmooth type-I at
u ∈ nR , if there exists η : X0 × nR → nR such that for every x ∈ X0,
f(x) − f(u) − f0(u; η) ∈ K
−g(u) − g0(u; η) ∈ Q.
Remark 2.4.1. If f : nR → R and g : nR → mR , K = R+, Q = mR+ then the
above definition reduces to that of type-I invex functions given by Sach et
al. [119].
Lemma 2.4.1. If (f, g) is (K×Q) generalized type-I at u ∈ nR with respect
to η : X0 × nR → nR then (f, g) is (K×Q)-nonsmooth type-I with respect to
same η.
Proof. Let (f, g) be (K×Q) generalized type-I at u ∈ nR , with respect to
η : X0 × nR → nR , then for every x ∈ nR , for all A ∈ ∂f(u), B∈∂g(u),
f(x) − f(u) − Aη(x, u) ∈ K
and −g(u) − Bη(x, u) ∈ Q.
For each index i ∈ {1, 2,...,m}, choose ( )i iv f u∈ ∂ such that
, sup{ , : ( )}i i i iv v v f u= ∈ ∂η η = 0( ; )if u η
then 1 2( , ,....., ) ( )mA v v v f u= ∈ ∂ and ( ) ( ) ( , )f x f u A x u K− − ∈η
which implies that,
f(x) − f(u) − f0(u; η) ∈ K.
91
Similarly, for each j ∈ {1, 2,...,p} we can choose ( )j jw g u∈ ∂ and obtain
1 2( , ,...., ) ( )pB w w w g u= ∈ ∂ .
Thus ( ) ( , )g u B x u Qη− − ∈ ,
which gives that −g(u) − g0(u; η) ∈ Q.
Hence (f, g) is (K×Q) non smooth type-I at u, with respect to same η.
The following example shows that the converse of the above lemma is not
true.
Example 2.4.1. Consider the problem
(VP0) K-minimize f (x)
subject to − g(x) ∈ Q
where f : R → 2R , g : R → 2R , f(x) = (f1(x), f2(x)), g(x) = (g1(x), g2(x)),
K = {(x, y) : x ≤ y, 0x ≤ } and Q = {(x, y): x ≥ y, 0y ≤ };
1
1, 0( )
1, 0x x
f xx
− <=
− ≥
3
2
, 0( )
0, 0x x x
f xx
− <=
≥
1
1 , 0( )
1, 0x x
g xx
+ <=
≥ 2 2
2, 0( )
2 , 0x
g xx x
<=
− ≥
then −g(x) ∈ Q ⇒ x ≤ 1.
Therefore the feasible set becomes X0 = {x ∈ R: x ≤ 1}.
Define η: X0 × R → R as η(x, u) = (x − u)3.
Now, 01 1
0, 0(0) [0,1], (0, )
, 0v
f f vv v
<∂ = =
≥
92
02 2
0, 0(0) [0,1], (0, )
, 0v
f f vv v
<∂ = =
≥
therefore
f(x) − f(0) − f0(0; η) ∈ K.
Further −g(0) − g0(0; η) ∈ Q, for every x∈X0.
Hence (f, g) is (K×Q)-nonsmooth type-I at u = 0.
However, for 1 1
1(0),
4A f= ∈ ∂ 2 2
1(0)
2A f= ∈ ∂ and x = 1 ∈ X0, A = (A1, A2)
1 1
( ) (0) ( ,0) ,4 2
f x f A x Kη− − − − = ∉
.
Thus, (f, g) is not (K×Q) generalized type-I at u = 0, with respect to η
defined above.
Remark 2.4.2. If f is K-generalized invex and g is Q-generalized invex at
u ∈ nR with respect to same η : X0 × nR → nR , then (f, g) is (K×Q)
generalized type-I at u. However the converse is not true as can be seen
from the following example.
Example 2.4.2. Consider the problem (VP0)
where 1 2
, 0( )
, 0x x
f xx x
<=
− ≥ 2
1 , 0( )
1, 0x x
f xx
+ <=
≥
2
1
3, 0( )
3, 0x x
g xx x
− + <=
+ ≥ 2
0, 0( )
, 02
xg x x
x
<=
− ≥
K = {(x, y): x ≤ 0, y ≤ −x} and Q = {(x, y) : x ≤ y, y ≥ 0}.
Then, −g(x) ∈ Q ⇒ x ≥ 3− . Hence X0 = { x ∈ R : x ≥ 3− }
93
Here ∂f1(0) = [0, 1], ∂f2(0) = [0, 1], ∂g1(0) = [0, 1], ∂g2(0) = [−1/2, 0].
Define η : X0 × R → R as η(x, u) = (x −u)2.
Then, (f, g) is (K×Q) generalized type-I at u = 0, with respect to η defined
above.
As, f(x) − f(0) − Aη(x, 0) ∈ K, for every x ∈ X0, A ∈ ∂f(0)
and −g(0) − Bη(x, 0) ∈ Q, for every x ∈ X0 and B ∈ ∂g(0).
Further, for 1 1 2 2
1 1(0), (0)
2 4B g B g
−= ∈ ∂ = ∈ ∂ and x = 1 ∈ X0,
g( x ) − g(0) − Bη(x, 0) = 1 1
,2 4
−
∉Q.
Therefore (f, g) is not (K×Q) generalized invex at u = 0.
Remark 2.4.3. If K = mR+ , Q = pR+ and (f, g) is (K×Q) generalized type-I
with respect to η then (f, g) is generalized type-I [79] with respect to
same η. However, the converse fails as can be viewed from the following
example.
Example 2.4.3. The pair ( f, g) considered in Example 2.4.2, has been
proved to be (K×Q) generalized type-I with respect to η(x, u) = (x − u)2.
However (f, g) is not generalized type-I with respect to same η, because,
for x = 1 ∈ X0, 1
1,
2A = f1(x) − f1(0) − A1η(x, 0) =
32
− < 0 and for B1 =
14
,
−g1(0) − B1η(x, 0) =134
− < 0.
Based on the lines of Soleimani-damaneh [125] we now introduce various
generalizations of (K×Q) generalized type-I functions.
94
Definition 2.4.3. (f, g) is said to be (K×Q) generalized pseudo type-I
at u ∈ nR if there exists η : X0 × nR → nR such that for all x ∈ X0, some
A ∈ ∂f(u), B ∈ ∂g(u),
−Aη(x, u) ∉ int K ⇒ −(f(x) − f(u)) ∉ int K.
−Bη(x, u) ∉ int Q ⇒ g(u) ∉ int Q.
In other words, (f, g) is said to be (K×Q) generalized pseudo-type-I at
u ∈ nR , if there exists η : X0 × nR → nR such that for every x ∈ X0 and for
all A ∈ ∂f(u), B ∈ ∂g(u),
− (f(x) − f(u)) ∈ int K ⇒ −Aη(x, u) ∈ int K
g(u) ∈ int Q ⇒ −Bη(x, u) ∈ int Q.
Definition 2.4.4. (f, g) is said to be (K×Q) generalized quasi type-I at
u ∈ nR , if there exists η : X0 × nR → nR such that for every x ∈ X0, for all
A ∈ ∂f(u), B ∈ ∂g(u)
f(x) − f(u) ∉ int K ⇒ −Aη(x, u) ∈ K
−g(u) ∉ int Q ⇒ −Bη(x, u) ∈ Q.
Definition 2.4.5. (f, g) is said to be (K×Q) generalized pseudo quasi
type-I at u ∈ nR , if there exists η : X0 × nR → nR such that for every
x ∈ X0, for all A ∈ ∂f(u), B ∈ ∂g(u)
−(f(x) − f(u)) ∈ int K ⇒ −Aη(x, u) ∈ int K
−g(u) ∉ int Q ⇒ −Bη(x, u) ∈ Q.
Definition 2.4.6. (f, g) is said to be (K×Q) generalized quasi pseudo
type-I at u ∈ nR , if there exists η : X0 × nR → nR such that for every
x ∈ X0 for all A ∈ ∂f(u) and some B∈∂g(u)
95
f(x) − f(u) ∉ int K ⇒ −Aη(x, u) ∈ K
−Bη(x, u) ∉ int Q ⇒ g(u) ∉ int Q.
Definition 2.4.7. (f, g) is said to be strictly (K×Q) generalized pseudo
quasi type-I at u ∈ nR , if there exists η : X0 × nR → nR , such that for
every x ∈ X0, for all A ∈ ∂f(u), B ∈ ∂g(u)
− (f(x) − f(u)) ∈ K ⇒ − Aη(x, u) ∈ int K
−g(u) ∉ int Q ⇒ −Bη(x, u) ∈ Q.
Definition 2.4.8. (f, g) is said to be strongly (K×Q) generalized
pseudo quasi type-I at u ∈ nR , if there exists η : X0 × nR → nR such that
for every x ∈ X0, A∈ ∂f(u)
−Aη(x, u) ∉ int K ⇒ f(x) − f(u) ∈ K,
−g(u) ∉ int Q ⇒ −Bη(x, u) ∈ Q, for all B ∈ ∂g(u).
We now give an example of a function which is (K×Q)-generalized pseudo
type-I but not (K×Q) generalized type-I.
Example 2.4.4. Consider the problem (VP0)
where 2
1 2 3
, 1, 1( ) ( )
, 1, 1
x xx xf x f x
x xx x
<− < = = ≥− ≥
1 2
, 1 0 1( ) ( )
1, 1 1, 1x x x
g x g xx x x
< <= =
≥ − + ≥
K = {(x, y) : y ≤ x, x ≥ 0}, Q = {(x, y) : y ≤ −x, 0y ≥ }
Now −g(x) ∈ Q ⇒ 0 ≤ x ≤ 2,
therefore the feasible set is X0 = {x ∈ R : 0 ≤ x ≤ 2}.
96
Here ∂f1(1) = [−2, −1], ∂f2(1) = {1}, ∂g1(1) = [0, 1] and ∂g2(1) = [−1, 0]
Define η : X0 × R → R as η(x, u) = x2 − u2
Then (f, g) is (K×Q) generalized pseudo type-I at u = 1, with respect to η,
because for A ∈ ∂f(1)
−Aη(x, 1) ∉ int K ⇒ 0 1x≤ ≤ ⇒ − (f(x) − f(1)) ∉ int K
and for B ∈ ∂g(1), −Bη(x, 1) ∉ int Q ⇒ g(1) ∉ int Q.
On the other hand,
for A = (−2, 1) and x = 0 ∈ X0,
( ) (1) ( ,1) ( 1,0)f x f A x K− − = − ∉η .
Hence (f, g) fails to be (K×Q) generalized type-I.
We now give an example to show that (K×Q) generalized pseudo quasi
type-I function may fail to be (K×Q) generalized type-I.
Example 2.4.5. Consider the problem (VP0)
where 1 2
1, 0 2, 0( ) ( )
1, 0 2, 0x x x
f x f xx x x
< − < = =
− + ≥ − ≥
1 22 3 2
, 0 0, 0( ) ( )
, 0 , 0x x x
g x g xx x x x x
< < = =
≥ − ≥
K = {(x, y) : x ≤ 0, y ≤ x}, Q = {(x, y) : x ≤ 0, y ≤ −x}.
Then, −g(x) ∈ Q ⇒ x ≥ 0, therefore the feasible set is X0 = {x ∈ R : x ≥ 0}.
Here ∂f1(0) = [−1, 0], ∂f2(0) = [0, 1], ∂g1(0) = [0, 1], ∂g2(0) = {0}
Define η : X0 × R → R as η(x, u) = x3 − u3.
97
Then (f, g) is (K×Q) generalized pseudo quasi type-I, at u = 0 with respect
to η because, for A ∈ ∂f(0)
−Aη(x, 0) ∉ int K ⇒ −(f(x) − f(0)) ∉ int K
and
−g(0) ∉ int Q ⇒ −Bη(x, 0) ∈ Q, for all x ∈ X0, B ∈ ∂g(0).
However, (f, g) fails to be (K×Q) generalized type-I at u = 0,
because for 1 1
,2 4
A = −
∈ ∂f(0) and x = 1 ∈ X0, we have
f(x) − f(0) − Aη(x, 0) = 1 1
,2 4
− −
∉ K.
Suneja et al. [137] gave the following necessary optimality conditions for
(VP).
Theorem 2.4.1 ([137], Fritz John type necessary optimality
conditions). Let f be K-generalized invex and g be Q-generalized invex
at x0 ∈ X0 with respect to same η : nR × nR → nR . If (VP) attains a weak
minimum at x0 then there exist λ ∈ K+, µ ∈ Q+ not both zero such that
0 ∈ ∂(λTf )(x0) + ∂(µTg)(x0) (2.53)
and
µTg(x0) = 0 (2.54)
The following constraint qualification is used to establish Karush-Kuhn-
Tucker type necessary optimality conditions.
Definition 2.4.9. The problem (VP) is said to satisfy generalized
Slater constraint qualification, if there exists x ∈ nR such that
− ( ) intg x Q∈ .
98
Theorem 2.4.2 ([137], Karush-Kuhn-Tucker type necessary
optimality conditions). Let f be K-generalized invex and g be
Q-generalized invex at x0 ∈ X0 with respect to same η : nR × nR → nR .
Suppose that generalized Slater constraint qualification is satisfied.
If (VP) attains a weak minimum at x0, then there exist 0 ≠ λ ∈ K + ,
µ ∈ Q+ such that (2.53) and (2.54) hold.
We now obtain sufficient optimality conditions for (VP).
Theorem 2.4.3. Let (f, g) be (K×Q) generalized type-I at x0 ∈ X0 with
respect to η : X0 × nR → nR and suppose that there exist 0 ≠ λ* ∈ K+,
µ* ∈ Q+, such that
0 ∈ ∂(λ*Tf)(x0) + ∂(µ*Tg)(x0) (2.55)
µ*Tg(x0) = 0 (2.56)
Then x0 is a weak minimum of (VP).
Proof. Let if possible x0 be not a weak minimum of (VP). Then there
exists x ∈ X0 such that
f(x0) − f(x) ∈ int K (2.57)
Since (f, g) is (K×Q)-generalized type-I at x0 ∈ X0, we have
f(x) − f(x0) − Aη(x, x0) ∈ K, for all A ∈ ∂f(x0) (2.58)
and
−g(x0) − Bη(x, x0) ∈ Q, for all B ∈ ∂g(x0) (2.59)
Adding (2.57) and (2.58) we get,
− Aη(x, x0) ∈ int K, for all A ∈ ∂f(x0). (2.60)
Since 0 ≠ λ* ∈ K+, we have from (2.60),
99
λ* Aη(x, x0) < 0, for all A ∈ ∂f(x0).
By virtue of (2.55), there exist x* ∈ ∂(λ*Tf)(x0), y* ∈ ∂(µ*Tg)(x0) such that
x* + y* = 0 (2.61)
As x* ∈ ∂(λ*Tf(x0)) = λ*∂f(x0), we get
x*η(x, x0) < 0.
If * 0µ ≠ , then we have − y*η(x, x0) < 0.
As y* ∈ ∂(µ*Tg)(x0) = µ* ∂g(x0), we get y* = µ*B*, for some B* ∈ ∂g(x0).
Thus,
−µ*B* η(x, x0) < 0, B* ∈ ∂g(x0). (2.62)
Now µ* ∈ Q+, therefore from (2.59) we have, − µ*T g(x0) − µ*Bη(x, x0) ≥ 0,
for all B ∈ ∂g(x0), which on using (2.56) gives
−µ*Bη(x, x0) ≥ 0, for all B ∈ ∂g(x0).
This is a contradiction to (2.62).
If * 0µ = , then also we arrive at a contradiction.
Hence x0 is a weak minimum of (VP). 1
Theorem 2.4.4. Let (f, g) be (K×Q) generalized pseudo quasi type-I at
x0 ∈ X0 with respect to η : X0 × nR → nR and suppose there exist
0 ≠ λ* ∈ K+, µ* ∈ Q+, such that (2.55), (2.56) hold, then x0 is a weak
minimum of (VP).
Proof. Let if possible, x0 be not a weak minimum of (VP) then there
exists x ∈ X0 such that
f(x0) − f(x) ∈ int K. (2.63)
100
As (2.55) holds, therefore there exist x* ∈ ∂(λ*Tf )(x0), y* ∈ ∂(µ*Tg)(x0) such
that (2.61) is satisfied.
Since (f, g) is (K×Q) generalized pseudo quasi type-I at x0 ∈ X0, therefore,
from (2.63) we have
−Aη(x, x0) ∈ int K, for all A ∈ ∂f(x0).
Now, 0 ≠ λ* ∈ K+, gives
λ*Aη(x, x0) < 0, for all A ∈ ∂f(x0)
which implies,
x*η(x, x0) < 0 as x* ∈ ∂(λ*Tf)(x0). (2.64)
From (2.56), µ*Tg(x0) = 0, which gives −g(x0) ∉ int Q, if * 0µ ≠ .
As (f, g) is (K×Q) generalized pseudo quasi type-I at x0
−Bη(x, x0) ∈ Q, for all B ∈ ∂g(x0),
which implies µ*Bη(x, x0) ≤ 0 as µ*∈Q+ .
The above inequality also holds for * 0µ = .
As y* ∈∂(µ*Tg)(x0), we have y*η(x, x0) ≤ 0,
which on using (2.61) gives x*η(x, x0) ≥ 0.
This contradicts (2.64).
Hence x0 is a weak minimum of (VP). ð
Theorem 2.4.5. Let (f, g) be (K×Q) generalized type-I at x0 ∈ X0 with
respect to η : X0 × nR → nR Suppose there exist λ* ∈ KS+, µ* ∈ Q+, such
that (2.55) and (2.56) hold. Then x0 is a minimum of (VP).
101
Proof. Let if possible, x0 be not a minimum of (VP), then there exists
x∈X0 such that
f(x0) − f(x) ∈ K \ {0} (2.65)
As (2.55) holds, there exist x* ∈ ∂(λ*Tf )(x0), y* ∈ ∂(µ*Tg)(x0) such that (2.61)
holds.
Since (f, g) is (K×Q) generalized type-I at x0 ∈ X0, therefore proceeding on
the similar lines as in proof of Theorem 2.4.3 and using (2.65) we have
−Aη(x, x0) ∈ K \ {0}.
As λ* ∈ KS+, we have
λ*Aη(x, x0) < 0, for all A ∈ ∂f(x0).
This leads to a contradiction as in Theorem 2.4.3. Hence x0 is a minimum
of (VP). 1
Theorem 2.4.6. Let (f, g) be strictly (K×Q) generalized pseudo quasi
type-I at x0 ∈ X0 with respect to η : X0 × nR → nR . Suppose that there
exist 0 ≠ λ*∈K+, µ* ∈ Q+, such that (2.55) and (2.56) hold. Then x0 is a
minimum of (VP).
Proof. It follows from (2.55) that there exist x* ∈ ∂(λ*Tf)(x0),
y* ∈ ∂(µ*Tg)(x0) such that (2.61) holds. Let if possible x0 be not a minimum
of (VP) then there exists x ∈ X0 such that (2.65) holds.
Since (f, g) is strictly (K×Q) generalized pseudo quasi type-I, therefore,
we have
− Aη(x, x0) ∈ int K, for all A ∈ ∂f(x0).
102
Proceeding on the same lines as in proof of Theorem 2.4.4, we arrive at a
contradiction. Hence x0 is a minimum of (VP). 1
Theorem 2.4.7. Let (f, g) be (K×Q) generalized type-I at x0 ∈ X0 with
respect to 0: n nX R Rη × → and suppose that there exist λ* ∈ K+, µ* ∈ Q+,
(λ*, µ* ) ≠ 0 such that (2.55) and (2.56) hold. Then x0 is a minimum
solution of the scalarized problem
*( )λ
VP minimize *( )( )T f xλ
subject to − g(x) ∈ Q
* sKλ +∈
and is hence a Benson proper minimizer of (VP).
The proof of the above theorem follows on the lines of Theorem 2.4.3. 1
Theorem 2.4.8. Let (f, g) be strongly (K×Q) generalized pseudo quasi
type-I at x0 ∈ X0 with respect to η : X0 × nR → nR and suppose that there
exist ≠0 λ* ∈ K+, µ* ∈ Q+ such that (2.55) and (2.56) hold. Then x0 is a
strong minimum of (VP).
Proof. By virtue of (2.55), there exist x* ∈ ∂(λ*Tf)(x0), y* ∈ ∂(µ*Tg)(x0) such
that (2.61) holds.
Let if possible x0 be not a strong minimum of (VP), then there exists x∈X0
such that f(x) − f(x0) ∉ K.
Since (f, g) is strongly (K×Q) generalized pseudo quasi type-I at x0, we
have
−Aη(x, x0) ∈ int K, for all A ∈ ∂f(x0)
Now proceeding as in Theorem 2.4.4 we arrive at a contradiction, thus
proving that x0 is a strong minimum of (VP). 1
103
We now consider the Mond-Weir type dual of the vector optimization
problem (VP).
(VD) K-maximize f(u)
subject to
0 ∈ ∂(λT f )(u) + ∂(µT g)(u) (2.66)
µTg(u) ≥ 0
0 ≠ λ∈K+, µ ∈ Q+
We now establish weak duality result for the pair (VP) and (VD).
Theorem 2.4.9 (Weak duality). Let x be feasible for (VP) and (u, λ, µ)
be feasible for (VD). Let (f, g) be (K×Q) generalized type-I at u with
respect to η : X0 × Rn → Rn then f(u) − f(x) ∉ int K.
Proof. Since (u, λ, µ) is feasible for (VD) therefore by (2.66) there exist
x* ∈ ∂(λT f )(u), y* ∈ ∂(µT g)(u) such that
x* + y* = 0. (2.67)
Let if possible f(u) − f(x) ∈ int K (2.68)
Now (f, g) is (K×Q) generalized type-I at u, therefore we have
f(x) − f(u) − Aη(x, u) ∈ K, for all A ∈ ∂f(u) (2.69)
− g(u) − Bη(x, u) ∈ Q for all B ∈ ∂g(u). (2.70)
Adding (2.68) and (2.69) we have
− Aη(x, u) ∈ int K, for all A ∈ ∂f(u).
As 0 ≠ λ ∈ K+, we get
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λAη(x, u) < 0
which implies
x*η(x, u) < 0, as x* ∈ ∂(λTf)(u).
By (2.67) we have −y*η(x, u) < 0.
Since y* ∈ ∂(µTg)(u), we obtain y* = µB*, B* ∈ ∂g(u).
Thus we have −µB*η(x, u) < 0 for 0µ ≠ . (2.71)
From (2.70) we get
−µTg(u) − µB*η(x, u) ≥ 0, as µ ∈ Q+
Since u is feasible for (VD), we get
−µB*η(x, u) ≥ 0,
which contradicts (2.71).
If 0µ = , then also we arrive at a contradiction.
Hence f(u) − f(x) ∉ int K. ð
Theorem 2.4.10 (Weak Duality). Let x ∈ X0 and (u, λ, µ) be feasible for
(VD), (f, g) be (K×Q) generalized pseudo quasi type-I at u with respect to
η : X0 × nR → nR and (λ, µ) ≠ 0 then f(u) − f(x) ∉ int K.
Proof. Since (u, λ, µ) is feasible for (VD), by (2.66), there exists
x* ∈ ∂(λT f )(u), y* ∈ ∂(µT g)(u) such that (2.67) holds.
Let if possible f(u) − f(x) ∈ int K.
Since (f, g) is (K×Q) generalized pseudo quasi type-I at u we get
−Aη(x, u) ∈ int K, for all A ∈ ∂f(u).
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Now proceeding as in Theorem 2.4.9, we get
− µ B*η(x, u) < 0. (2.72)
Also, (u, λ, µ) is feasible for (VD), so we have
µTg(u) ≥ 0
⇒ −g(u) ∉ int Q if 0µ ≠
⇒ −Bη(x, u) ∈ Q for all B ∈ ∂g(u), as (f, g) is (K×Q) generalized
pseudo quasi type-I at u.
⇒ in particular −µB*η(x, u) ≥ 0.
The above inequality also holds for 0µ = .
But this contradicts (2.72), hence f(u) − f(x) ∉ int K. ð
We next prove strong duality result.
Theorem 2.4.11 (Strong Duality). Suppose that (VP) attains a weak
minimum at x0 and generalized Slater constraint qualification is
satisfied. Let ( f, g) be (K×Q) generalized invex at x0 with respect to
η : nR × nR → nR , then there exist 0 ≠ λ0 ∈ K + , µ0 ∈ Q+ such that
( 0x , λ0, µ0) is a feasible solution of (VD). Further if the conditions of
Weak Duality Theorem 2.4.10 hold for all feasible solution of (VP) and of
(VD), then ( 0x , λ0, µ0) is a weak maximum of (VD).
Proof. As x0 is a weak minimum of (VP), by Theorem 2.4.2, there exist
0 ≠ λ0 ∈ K + , µ0 ∈ Q+ such that
0 ∈ ∂( 0Tλ f )(x0) + ∂( 0
Tµ g)(x0)
and 0Tµ g(x0) = 0,
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which clearly shows that ( 0x , λ0, µ0) is a feasible solution of (VD) and the
values of two objectives functions are equal. Further if ( 0x , λ0, µ0) is not a
weak maximum of (VD), then there exists a feasible solution (u, λ, µ) of
(VD) such that f(u) − f( 0x ) ∈ int K, which contradicts Theorem 2.4.10.
Hence ( 0x , λ0, µ0) is a weak maximum of (VD). ð
