# On a Nonsmooth Vector Optimization Problem with Generalized Cone Invexity

Authors: Hehua Jiao; Sanyang Liu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

By using Clarke’s generalized gradients we consider a nonsmooth vector optimization problem with cone constraints and introduce some generalized cone-invex functions calledK-α-generalized invex, K-α-nonsmooth invex, and other related functions. Several sufficient optimality conditions and Mond-Weir type weak and converse duality results are obtained for this problem under the assumptions of the generalized cone invexity. The results presented in this paper generalize and extend the previously known results in this area.


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## Body

## 1. Introduction

In optimization theory, convexity plays a key role in many aspects of mathematical programming including sufficient optimality conditions and duality theorems; see [1, 2]. Many attempts have been made during the past several decades to relax convexity requirement; see [3–7]. In this endeavor, Hanson [8] introduced invex functions and studied some applications to optimization problem. Subsequently, many authors further weakened invexity hypotheses to establish optimality conditions and duality results for various mathematical programming problems; see, for example, [9–11] and the references cited therein.Above all, Yen and Sach [12] introduced cone-generalized invex and cone-nonsmooth invex functions. Giorgi and Guerraggio [13] presented the notions of α-K-invex, α-K pseudo-invex, and α-K quasi-invex functions in the differentiable case and derived optimality and duality results for a vector optimization problem over cones. Khurana[14] extended pseudoinvex functions to differentiable cone-pseudoinvex and strongly cone-pseudoinvex functions. Based on this, Suneja et al. [15] defined cone-nonsmooth quasi-invex, cone-nonsmooth pseudoinvex, and other related functions in terms of Clarke’s [16] generalized directional derivatives and established optimality and duality results for a nonsmooth vector optimization problem.On the other hand, Noor [17] proposed several classes of α-invex functions and investigated some properties of the α-preinvex functions and their differentials. Mishra et al. [18] defined strict pseudo-α-invex and quasi-α-invex functions. Mishra et al. [19] further introduced the concepts of nonsmooth pseudo-α-invex functions and established a relationship between vector variational-like inequality and nonsmooth vector optimization problems by using the nonsmooth α-invexity.In the present paper, by using Clarke's generalized gradients of locally Lipschitz functions we are concerned with a nonsmooth vector optimization problem with cone constraints and introduce several generalized invex functions over cones namelyK-α-generalized invex, K-α-nonsmooth invex, and other related functions, which, respectively, extend some corresponding concepts of [12, 13, 15, 17]. Some sufficient optimality conditions for this problem are obtained by using the above defined concepts. Furthermore, a Mond-Weir type dual is formulated and a few weak and converse duality results are established. We generalize and extend some results presented in the literatures on this topic.


## 2. Preliminaries and Definitions

Throughout this paper, letη:Rn×Rn→Rn and α:Rn×Rn→R+∖{0} be two fixed mappings. intK and K- denote the interior and closure of K⊆Rm, respectively. We always assume that K is a closed convex cone with intK≠∅.The positive dual coneK+ of K is defined as
(2.1)K+={x*∈Rm:〈x*,x〉≥0,∀x∈K}.
The strict positive dual cone K++ of K is given by
(2.2)K++={x*∈Rm:〈x*,x〉>0,∀x∈K∖{0}}.
The following property is from [20], which will be used in the sequel.Lemma 2.1 (see [20]).
LetK⊆Rm be a convex cone with intK≠∅. Then,(a)
∀u*∈K+∖{0},x∈intK⇒〈u*,x〉>0;(b)
∀u*∈intK+,x∈K∖{0}⇒〈u*,x〉>0.A functionψ:Rn→R is called locally Lipschitz at u∈Rn, if there exists l>0 such that
(2.3)∥ψ(x)-ψ(y)∥≤l∥x-y∥,
for all x,y in a neighbourhood of u.A functionψ is called locally Lipschitz on Rn, if it is locally Lipschitz at each point of Rn.Definition 2.2 (see [16]).
Letψ:Rn→R be a locally Lipschitz function, then ψ∘(u;v) denotes Clarke's generalized directional derivative of ψ at u∈Rn in the direction v and is defined as
(2.4)ψ∘(u;v)=limsupy→ut→0ψ(y+tv)-ψ(y)t.
Clarke's generalized gradient of ψ at u is denoted by ∂ψ(u) and is defined as
(2.5)∂ψ(u)={ξ∈Rn∣ψ∘(u;v)≥〈ξ,v〉,∀v∈Rn}.
Let f:Rn→Rm be a vector-valued function given by f=(f1,f2,…,fm), where fi:Rn→R,i=1,2,…,m. Then f is said to be locally Lipschitz on Rn if each fi is locally Lipschitz on Rn. The generalized directional derivative of a locally Lipschitz function f:Rn→Rm at u∈Rn in the direction v is given by
(2.6)f∘(u;v)={f1∘(u;v),f2∘(u;v),…,fm∘(u;v)}.
The generalized gradient of f at u is the set
(2.7)∂f(u)=∂f1(u)×∂f2(u)×⋯×∂fm(u),
where ∂fi(u)(i=1,2,…,m) is the generalized gradient of fi at u.
EveryA=(a1,a2,…,am)∈∂f(u) is a continuous linear operator from Rn to Rm and
(2.8)Au=(〈a1,u〉,〈a2,u〉,…,〈am,u〉)∈Rm,∀u∈Rn.Lemma 2.3 (see [16]).
(a) Iffi:Rn→R is locally Lipschitz then, for each u∈Rn,
(2.9)fi∘(u;v)=max{〈ξ,v〉∣ξ∈∂fi(u)},∀v∈Rn,i=1,2,…,m.
(b) Letfi(i=1,2,…,m) be a finite family of locally Lipschitz functions on Rn, then ∑i=1m‍fi is also locally Lipschitz and
(2.10)∂(∑i=1m‍fi)(u)⊆∑i=1m‍∂fi(u),∀u∈Rn.Definition 2.4 (see [17]).
A functionh:Rn→R is said to be α-invex function at u∈Rn with respect to α and η, if there exist functions α and η such that, for every x∈Rn, we have
(2.11)h(x)-h(u)≥〈α(x,u)∇h(u),η(x,u)〉.
In this paper, we consider the following vector optimization problem with cone constraints:
(VP)K-minf(x)s.t.-g(x)∈Q,
where f:Rn→Rm,g:Rn→Rp are locally Lipschitz functions on Rn and K,Q are closed convex cones with nonempty interiors in Rm and Rp, respectively.
DenoteX={x∈Rn:-g(x)∈Q} the feasible set of problem (VP).
For eachλ∈K+ and μ∈Q+, we suppose that λf=λ∘f and μg=μ∘g are locally Lipschitz.
Now, we present the concepts of solutions for problem(VP) in the following sense.Definition 2.5.
Letu∈X, then (a)
u is said to be a minimum of (VP) if for all x∈X,
(2.12)f(u)-f(x)∉K∖{0};(b)
u is said to be a weak minimum of (VP) if for all x∈X,
(2.13)f(u)-f(x)∉intK;(c)
u is said to be a strong minimum of (VP) if for all x∈X,
(2.14)f(x)-f(u)∈K.Based on the lines of Yen and Sach [12] and Noor [17], we define the notions as follows.Definition 2.6.
Letf:Rn→Rm be a locally Lipschitz function. f is said to be K-α-generalized invex at u∈Rn, if there exist functions α and η such that for every x∈Rn and A∈∂f(u),
(2.15)f(x)-f(u)-α(x,u)Aη(x,u)∈K.Definition 2.7.
Letf:Rn→Rm be a locally Lipschitz function. f is said to be K-α-nonsmooth invex at u∈Rn, if there exist functions α and η such that for every x∈Rn,
(2.16)f(x)-f(u)-α(x,u)f∘(u;η)∈K,
where
(2.17)f∘(u;η)={f1∘(u;η),f2∘(u;η),…,fm∘(u;η)}.Remark 2.8.
Ifm=1,K=R+, and f is differentiable, then K-α-generalized invex and K-α-nonsmooth invex functions become α-invex function [17]; if α(x,u)≡1 for all x,u∈Rn, then K-α-generalized invex and K-α-nonsmooth invex functions reduce to K-generalized invex and K-nonsmooth invex functions defined by Yen and Sach [12].Lemma 2.9.
Iff is K-α-generalized invex at u with respect to α and η, then f is K-α-nonsmooth invex at u with respect to the same α and η.Proof.
Sincef is K-α-generalized invex at u, then there exist α and η such that for every x∈Rn and A∈∂f(u)(2.18)f(x)-f(u)-α(x,u)Aη(x,u)∈K.
By Lemma 2.3, for each i∈{1,2,…,m}, we choose a~i∈∂fi(u) such that
(2.19)〈a~i,η〉=max{〈ai,η〉:ai∈∂fi(u)}=fi∘(u;η).
Then A~=(a~1,…,a~m)∈∂f(u) and
(2.20)f(x)-f(u)-α(x,u)A~η(x,u)∈K,
equivalently,
(2.21)f(x)-f(u)-α(x,u)f∘(u;η)∈K.
Hence, f is K-α-nonsmooth invex at u with respect to the same α and η.The following example shows that converse of the above lemma is not true.Example 2.10.
LetK={(x,y)∣y≤-x} be a cone in R2. Assume that f=(f1,f2), where
(2.22)f1(x)={2-x2,x≥0,2,x<0,f2(x)={-x+1,x≥0,1,x<0.
Let α:R×R→R+∖{0} and η:R×R→R be defined as α(x,u)=1/2 and η(x,u)=2x2-u3, respectively. Then at u=0,
(2.23)f(x)-f(0)-12f∘(0;η(x,0))={(-x2,-x+x2),x≥0,(0,0),x<0,∈K.
Hence, f is K-1/2-nonsmooth invex at u=0.
It is easy to verify∂f(0)={(0,v)∣-1≤v≤0}.
TakingA=(0,-1/4)∈∂f(0) and x=-1, we have
(2.24)f(-1)-f(0)-12Aη(-1,0)=(0,14)∉K.
Therefore, f is not K-1/2-generalized invex at u=0.
Next, we introduce several related functions ofK-α-nonsmooth invex.Definition 2.11.
f is said to be K-α-nonsmooth quasi-invex at u∈Rn, if there exist functions α and η such that for every x∈Rn,
(2.25)f(x)-f(u)∉intK⇒-α(x,u)f∘(u;η(x,u))∈K.Definition 2.12.
f is said to be K-α-nonsmooth pseudo-invex at u∈Rn, if there exist functions α and η such that for every x∈Rn,
(2.26)f(u)-f(x)∈intK⇒-α(x,u)f∘(u;η(x,u))∈intK.Definition 2.13.
f is said to be strict K-α-nonsmooth pseudo-invex at u∈Rn, if there exist functions α and η such that for every x∈Rn,
(2.27)f(u)-f(x)∈K⇒-α(x,u)f∘(u;η(x,u))∈intK.Definition 2.14.
f is said to be strong K-α-nonsmooth pseudo-invex at u∈Rn, if there exist functions α and η such that for every x∈Rn,
(2.28)f(x)-f(u)∉K⇒-α(x,u)f∘(u;η(x,u))∈intK.Remark 2.15.
Ifα(x,u)≡1 for all x,u∈Rn and f is differentiable, then K-α-nonsmooth pseudo-invex and strong K-α-nonsmooth pseudo-invex functions reduce to K-pseudo-invex and strong K-pseudo-invex functions, defined by Khurana [14].Remark 2.16.
Ifm=1,K=R+,α(x,u)≡1 for all x,u∈Rn, and f is differentiable, then K-α-nonsmooth quasi-invex functions reduce to quasi-invex functions and K-α-nonsmooth pseudo-invex and strong K-α-nonsmooth pseudo-invex functions reduce to pseudo-invex functions [8].Remark 2.17.
Ifα(x,u)≡1 for all x,u∈Rn, then the above definitions reduce to the corresponding definitions [15]. If f is differentiable, then K-α-generalized invex and K-α-nonsmooth pseudo-invex functions reduce to α-K-invex and α-K pseudo-invex functions [13], respectively.


## 3. Optimality Criteria

In this section, we establish a few sufficient optimality conditions for problem(VP) by using the above defined functions.Theorem 3.1.
Letf be K-α-generalized invex and g be Q-α-generalized invex at u∈X with respect to the same α and η. We assume that there exist λ∈K+,λ≠0,μ∈Q+ such that
(3.1)0∈∂(λf)(u)+∂(μg)(u),(3.2)μg(u)=0.
Then u is a weak minimum of (VP).Proof.
By contradiction, we assume thatu is not a weak minimum of (VP). Then there exists a feasible solution x of (VP) such that
(3.3)f(u)-f(x)∈intK.
From (3.1), it follows that there exist s∈∂(λf)(u) and t∈∂(μg)(u) such that
(3.4)s+t=0.
Since f is K-α-generalized invex and g is Q-α-generalized invex at u, we get
(3.5)f(x)-f(u)-α(x,u)Aη(x,u)∈K,∀A∈∂f(u),(3.6)g(x)-g(u)-α(x,u)Bη(x,u)∈Q,∀B∈∂g(u).
Summing (3.3) and (3.5), we have
(3.7)-α(x,u)Aη(x,u)∈intK,∀A∈∂f(u).
As λ∈K+,λ≠0,  from Lemma 2.1, we obtain
(3.8)α(x,u)λAη(x,u)<0,∀A∈∂f(u),
which yields
(3.9)α(x,u)sη(x,u)<0,asλ≠0,s∈∂(λf)(u)=λ∂f(u).
Considering positivity of α(x,u) and (3.4), one has
(3.10)tη(x,u)>0.
From t∈∂(μg)(u)=μ∂g(u), we deduce
(3.11)t=μB*,forsomeB*∈∂g(u).
Hence,
(3.12)μB*η(x,u)>0,whereB*∈∂g(u).
By μ∈Q+, relation (3.6) gives
(3.13)μg(x)-μg(u)-μα(x,u)Bη(x,u)≥0,∀B∈∂g(u).
By virtue of (3.2) and x∈X, the above inequality implies
(3.14)-μα(x,u)Bη(x,u)≥0,∀B∈∂g(u),
that is,
(3.15)μBη(x,u)≤0,∀B∈∂g(u),
which is a contradiction to (3.12).
Therefore,u is a weak minimum of (VP).Theorem 3.2.
Letf be K-α-generalized invex and g be Q-α-generalized invex at u∈X with respect to the same α and η. We assume that there exist λ∈K++,μ∈Q+ such that (3.1) and (3.2) hold. Then u is a minimum of (VP).Proof.
Assume contrary to the result thatu is not a minimum of (VP). Then there exists x∈X such that
(3.16)f(u)-f(x)∈K∖{0}.
From (3.1), it follows that there exist s∈∂(λf)(u) and t∈∂(μg)(u) such that
(3.17)s+t=0.
Since f is K-α-generalized invex at u∈X, we get
(3.18)f(x)-f(u)-α(x,u)Aη(x,u)∈K,∀A∈∂f(u).
Utilizing (3.16), we deduce
(3.19)-α(x,u)Aη(x,u)∈K∖{0},∀A∈∂f(u).
According to λ∈K++, we obtain
(3.20)α(x,u)λAη(x,u)<0,∀A∈∂f(u).
Next proceeding on the same lines as in the proof of Theorem 3.1, we obtain a contradiction.
Thus,u is a minimum of (VP).Theorem 3.3.
Letf be K-α-nonsmooth pseudo-invex and g be Q-α-nonsmooth quasi-invex at u∈X with respect to the same α and η. We assume that there exist λ∈K+,λ≠0,μ∈Q+ such that (3.1) and (3.2) hold. Then u is a weak minimum of (VP).Proof.
It follows from (3.1) that there exist s∈∂(λf)(u) and t∈∂(μg)(u) such that
(3.21)s+t=0.
Suppose that u is not a weak minimum of (VP). Then there exists x∈X such that
(3.22)f(u)-f(x)∈intK.
Since f is K-α-nonsmooth pseudo-invex at u∈X, we deduce
(3.23)-α(x,u)f∘(u;η(x,u))∈intK.
By λ∈K+,λ≠0 and Lemma 2.1, we obtain
(3.24)α(x,u)λf∘(u;η(x,u))<0.
From α(x,u)>0 and fi∘(u;η(x,u))=max{〈vi,η〉:vi∈∂fi(u)},i=1,2,…,m, we have
(3.25)λAη(x,u)<0,∀A∈∂f(u),
which implies,
(3.26)sη(x,u)<0,wheres∈∂(λf)(u)=λ∂f(u).
By x∈X and μ∈Q+,  -g(x)∈Q gives
(3.27)μg(x)≤0.
Taking (3.2) into account, one has
(3.28)μ(g(x)-g(u))≤0.
Next we prove
(3.29)μg∘(u;η(x,u))≤0.
If μ=0, inequality (3.29) holds obviously.
Ifμ≠0, from (3.28) and Lemma 2.1, we deduce
(3.30)g(x)-g(u)∉intQ.
Since g is Q-α-nonsmooth quasi-invex at u∈X, we have
(3.31)-α(x,u)g∘(u;η(x,u))∈Q.
From α(x,u)>0 and μ∈Q+, it follows that (3.29) also holds.
Similarly, by Lemma2.3, inequality (3.29) gives
(3.32)μBη(x,u)≤0,∀B∈∂g(u),
which yields,
(3.33)tη(x,u)≤0,wheret∈∂(μg)(u)=μ∂g(u).
Hence,
(3.34)sη(x,u)≥0,
which is in contradiction with (3.26).
Therefore,u is a weak minimum of (VP).The following example illustrates the above theorem.Example 3.4.
Consider the vector optimization problem(VP) where K={(x,y)y≥-x,y≥x},Q={(x,y)-x≤y≤x,x≥0}, and fi,gi:R→R,i=1,2 are defined as
(3.35)f1(x)={-x2-x,x>0,0,x≤0,f2(x)={-2x+1,x>0,1,x≤0,g1(x)={x3,x>0,x3,x≤0,g2(x)={x2,x>0,x2,x≤0.
Let α:R×R→R+∖{0} and η:R×R→R be defined as α(x,u)=2 and η(x,u)=(x+2u)3, respectively. It is easily testified that f and g are K-α-nonsmooth pseudo-invex and K-α-nonsmooth quasi-invex at u=0, respectively. The feasible set of (VP) is given by X=(-∞,0].
It is also easy to verify∂f(0)=[-1,0]×[-2,0],∂g(0)=[0,1/3]×[0,1/2].
Takingλ=(1,3)∈K+ and μ=(3,1)∈Q+, we have
(3.36)0∈[-7,32]=∂(λf)(0)+∂(μg)(0),μg(0)=0,
which imply that (3.1) and (3.2) hold.
Therefore, by Theorem3.3, u=0 is a weak minimum of (VP).Theorem 3.5.
Letf be strong K-α-nonsmooth pseudo-invex and g be Q-α-nonsmooth quasi-invex at u∈X with respect to the same α and η. We assume that there exist λ∈K+,λ≠0,μ∈Q+ such that (3.1) and (3.2) hold. Then u is a strong minimum of (VP).Proof.
From (3.1), it follows that there exist s∈∂(λf)(u) and t∈∂(μg)(u) such that
(3.37)s+t=0.
Assume that u is not a strong minimum of (VP). Then there exists x∈X such that
(3.38)f(x)-f(u)∉K.
Since f is strong K-α-nonsmooth pseudo-invex at u, we deduce
(3.39)-α(x,u)f∘(u;η(x,u))∈intK.
Next proceeding on the same lines as in the proof of Theorem 3.3, we get a contradiction.
Henceu is a strong minimum of (VP).Theorem 3.6.
Letf be strict K-α-nonsmooth pseudo-invex and g be Q-α-nonsmooth quasi-invex at u∈X with respect to the same α and η. We assume that there exist λ∈K+,λ≠0,μ∈Q+ such that (3.1) and (3.2) hold. Then u is a minimum of (VP).Proof.
From (3.1), it follows that there exist s∈∂(λf)(u) and t∈∂(μg)(u) such that
(3.40)s+t=0.
By contradiction, assume that u is not a minimum of (VP). Then there exists x∈X such that
(3.41)f(u)-f(x)∈K∖{0}⊂K.
Since f is strict K-α-nonsmooth pseudo-invex at u, we have
(3.42)-α(x,u)f∘(u;η(x,u))∈intK.
Next as in Theorem 3.3 we arrive at a contradiction.
Therefore,u is a minimum of (VP).


## 4. Duality

In relation to(VP), we consider the following Mond-Weir type dual problem:
(VD)K-maxf(y)s.t.0∈∂(λf)(y)+∂(μg)(y),μg(y)≥0,y∈Rn,λ∈K+,λ≠0,μ∈Q+.
Denote the feasible set of problem (VD) by G,  namely,  G={(y,λ,μ):0∈∂(λf)(y)+∂(μg)(y),μg(y)≥0,y∈Rn,λ∈K+,λ≠0,μ∈Q+}.Now, we establish weak and converse duality results.Theorem 4.1 (Weak duality).
Letx∈X and (y,λ,μ)∈G. If f is K-α-nonsmooth pseudo-invex and g is Q-α-nonsmooth quasi-invex at y with respect to the same α and η, then
(4.1)f(y)-f(x)∉intK.Proof.
Since(y,λ,μ)∈G, from (VD), it follows that there exist s∈∂(λf)(y) and t∈∂(μg)(y) such that
(4.2)s+t=0.
By contradiction, we assume that f(y)-f(x)∈intK.
Sincef is K-α-nonsmooth pseudo-invex at y, we have
(4.3)-α(x,y)f∘(y;η(x,y))∈intK.
By λ∈K+,λ≠0 and Lemma 2.1, we get
(4.4)α(x,y)λf∘(y;η(x,y))<0.
From α(x,y)>0 and Lemma 2.3, we deduce
(4.5)λAη(x,y)<0,∀A∈∂f(y),
which yields
(4.6)sη(x,y)<0,wheres∈∂(λf)(y)=λ∂f(y).
Using (4.2), we obtain
(4.7)tη(x,y)>0.
From t∈∂(μg)(y)=μ∂g(y), it follows that there exists B*∈∂g(y) such that
(4.8)t=μB*.
Hence,
(4.9)μB*η(x,y)>0,whereB*∈∂g(y).
From x∈X and (y,λ,μ)∈G, we find that
(4.10)μg(x)≤0≤μg(y).
Now we claim that
(4.11)μg∘(y;η(x,y))≤0.
Ifμ=0, then (4.11) holds trivially.
Ifμ≠0, from (4.10) and Lemma 2.1 we have
(4.12)g(x)-g(y)∉intQ.
As g is Q-α-nonsmooth quasi-invex at y, we obtain
(4.13)-α(x,y)g∘(y;η(x,y))∈Q,
which means that (4.11) also holds and is equivalent to
(4.14)μBη(x,y)≤0,∀B∈∂g(y),
which is a contradiction to (4.9). Thus
(4.15)f(y)-f(x)∉intK.Theorem 4.2 (Weak duality).
Letx∈X and (y,λ,μ)∈G. If f is K-α-generalized invex and g is Q-α-generalized invex at y with respect to the same α and η, then
(4.16)f(y)-f(x)∉intK.Proof.
Since(y,λ,μ)∈G, from (VD), it follows that there exist s∈∂(λf)(y) and t∈∂(μg)(y) such that
(4.17)s+t=0.
We assume contrary to the result that
(4.18)f(y)-f(x)∈intK.
Since f is K-α-generalized invex and g is Q-α-generalized invex at y, we get
(4.19)f(x)-f(y)-α(x,y)Aη(x,y)∈K,∀A∈∂f(y),(4.20)g(x)-g(y)-α(x,y)Bη(x,y)∈Q,∀B∈∂g(y).
Summing (4.18) and (4.19), we have
(4.21)-α(x,y)Aη(x,y)∈intK,∀A∈∂f(y).
By λ∈K+,λ≠0 and Lemma 2.1, we obtain
(4.22)α(x,y)λAη(x,y)<0,
which yields
(4.23)α(x,y)sη(x,y)<0,wheres∈∂(λf)(y)=λ∂f(y),λ≠0.
Applying positivity of α(x,y) and (4.17), we get
(4.24)tη(x,y)>0.
By the fact that t∈∂(μg)(y)=μ∂g(y), one has
(4.25)t=μB*,whereB*∈∂g(y).
Hence,
(4.26)μB*η(x,y)>0,whereB*∈∂g(y).
From μ∈Q+ and (4.20), we obtain
(4.27)μg(x)-μg(y)-μα(x,y)Bη(x,y)≥0,∀B∈∂g(y).
As x∈X and (y,λ,μ)∈G, we get
(4.28)μg(x)≤0≤μg(y).
Using the above relation, (4.27) yields
(4.29)-μα(x,y)Bη(x,y)≥0,∀B∈∂g(y),
that is,
(4.30)μBη(x,y)≤0,∀B∈∂g(y),
which contradicts (4.26). Therefore,
(4.31)f(y)-f(x)∉intK.Theorem 4.3 (Converse duality).
Lety∈X and (y,λ,μ)∈G. Assume that f is K-α-nonsmooth pseud-invex and g is Q-α-nonsmooth quasi-invex at y with respect to the same α and η. Then y is a weak minimum of (VP).Proof.
Since(y,λ,μ)∈G, from (VD), it follows that there exist s∈∂(λf)(y) and t∈∂(μg)(y) such that
(4.32)s+t=0.
Assume contrary to the result that y is not a weak minimum of (VP). Then there exists x-∈X such that
(4.33)f(y)-f(x-)∈intK.
Since f is K-α-nonsmooth pseudo-invex at y, we have
(4.34)-α(x-,y)f∘(y;η(x-,y))∈intK.
By λ∈K+,λ≠0 and Lemma 2.1, we get
(4.35)α(x-,y)λf∘(y;η(x-,y))<0,
which is equivalent to
(4.36)λAη(x-,y)<0,∀A∈∂f(y),
which yields
(4.37)sη(x-,y)<0,wheres∈∂(λf)(y)=λ∂f(y),λ≠0.
Using (4.32), we obtain
(4.38)tη(x-,y)>0.
As t∈∂(μg)(y)=μ∂g(y), thus t=μB*, where B*∈∂g(y).
Hence,(4.39)μB*η(x-,y)>0,whereB*∈∂g(y).
By x-∈X and (y,λ,μ)∈G, we have
(4.40)μg(x-)≤0≤μg(y).
By the similar argument to that of Theorem 4.1, we can prove that
(4.41)μg∘(y;η(x-,y))≤0,
which is equivalent to
(4.42)μBη(x-,y)≤0,∀B∈∂g(y),
which is in contradiction with (4.39).
Therefore,y is a weak minimum of (VP).Theorem 4.4 (Converse duality).
Lety∈X and (y,λ,μ)∈G. Assume that f is K-α-generalized invex and g is Q-α-generalized invex at y with respect to the same α and η. Then y is a weak minimum of (VP).Proof.
The proof of the above theorem is very similar to the proof of Theorem3.1, except that for this case we use the feasibility of (y,λ,μ) for (VD) instead of the relations (3.1) and (3.2).



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*Source: 458983-2012-11-18.xml*