Second order composed contingent epiderivatives and set valued vector equilibrium problems

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R E S E A R C H

Open Access

Second-order composed contingent

epiderivatives and set-valued vector

equilibrium problems

Qilin Wang

1*

_{, Zhi Lin}

1

_{, Jing Zeng}

2

_{, Zaiyun Peng}

1

_{and Xiaobing Li}

1

*_{Correspondence:}

[email protected]

1_{College of Sciences, Chongqing}

Jiaotong University, Chongqing, 400074, China

Full list of author information is available at the end of the article

Abstract

In this paper, we introduce the concept of a second-order composed contingent epiderivative for set-valued maps and discuss some of its properties. Then, by virtue of the second-order composed contingent epiderivative, we establish second-order suﬃcient optimality conditions and necessary optimality conditions for the weakly eﬃcient solution of set-valued vector equilibrium problems with unconstraints and set-valued vector equilibrium problems with constraints, respectively.

MSC: 90C46; 91B50

Keywords: set-valued vector equilibrium problems; second-order composed contingent epiderivatives; weakly eﬃcient solutions; second-order optimality conditions

1 Introduction

The vector equilibrium problem, which contains vector optimization problems, vector variational inequality problems and vector complementarity problems as special case, has been studied (see [–]). But so far, most papers focused mainly on the existence of so-lutions and the properties of the soso-lutions, there are a few papers which deal with the optimality conditions. Giannessiet al.[] turned the vector variational inequalities with constraints into another vector variational inequalities without constraints. They gave the sufficient conditions for the efficient solution and the weakly efficient solution of the vec-tor variational inequalities in finite dimensional spaces. Morgan and Romaniello [] gave the scalarization and Kuhn-Tucker-like conditions for weak vector generalized quasivari-ational inequalities in Hilbert space by using the subdifferential of the function. Gong [] presented the necessary and sufficient conditions for the weakly efficient solution, the Henig efficient solution and the superefficient solution for vector equilibrium problems with constraints under the condition of cone-convexity. Qiu [] presented the necessary and sufficient conditions for globally efficient solutions of vector equilibrium problems under generalized cone-subconvexlikeness. Gong and Xiong [] weakened the convexity assumption in [] and obtained the necessary and sufficient conditions for weakly effi-cient solutions of vector equilibrium problems. Under the nearly cone-subconvexlikeness, Longet al.[] obtained the necessary and sufficient conditions for the Henig efficient so-lution and the superefficient soso-lution to the vector equilibrium problems with constraints. By using the concept of Fréchet differentiability of mapping, Wei and Gong [] obtained

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the Kuhn-Tucker optimality conditions for weakly efficient solutions, Henig efficient so-lutions, superefficient solutions and globally efficient solutions to the vector equilibrium problems with constraints. Ma and Gong [] obtained the first-order necessary and suf-ficient conditions for the weakly efficient solution, the Henig efficient solution, and the globally proper efficient solution to the vector equilibrium problems with constraints.

It is well known that the second-order tangent sets and higher-order tangent sets in-troduced in [], in general, are not cones and convex sets, there are some diﬃculties in studying second-order and higher-order optimality conditions for general set-valued optimization problems. Until now, there are some papers to deal with higher-order opti-mality conditions by virtue of the higher-order derivatives or epiderivatives introduced by the higher-order tangent sets (see [–]). However, as far as we know the second-order optimality conditions of the solutions remain unstudied in set-valued vector equilibrium problems.

Motivated by the work reported in [, , –, ], we introduce a new second-order derivative called second-second-order composed contingent epiderivative for set-valued maps and obtain some of its properties. By virtue of the second-order composed contin-gent epiderivative, we obtain second-order suﬃcient optimality conditions and necessary optimality conditions for the weakly eﬃcient solution of set-valued vector equilibrium problems.

The rest of the paper is organized as follows. In Section , we recall some notions. In Section , we introduce second-order composed contingent epiderivatives for set-valued maps and discuss some of its properties. In Section , we establish second-order neces-sary and suﬃcient optimality conditions for weakly eﬃcient solutions to set-valued vector equilibrium problems.

2 Preliminaries and notations

Throughout this paper, letX,Y, andZ be three real normed spaces,Y∗ andZ∗ be the topological dual spaces ofY andZ, respectively. X, Y and Z denote the origins ofX,

Y, andZ, respectively. LetC⊂YandD⊂Zbe closed convex pointed cones inYandZ, respectively. LetMbe a nonempty subset inY. The cone hull ofMis deﬁned bycone(M) =

{ty|t≥,y∈M}. LetC∗be the dual cone ofconeC, deﬁned by

C∗=y∗∈Y∗:y∗(c)≥, for allc∈C.

LetEbe a nonempty subset ofX,G:E→Z_{be a set-valued map. The domain, the graph}

and the epigraph ofGare deﬁned, respectively, by

dom(G) =x∈E|G(x)=∅,

graph(G) =(x,z)∈X×Z|x∈E,z∈G(x),

epi(G) =(x,z)∈X×Z|x∈E,z∈G(x) +C.

Denote

G(E) =

x∈E

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Deﬁnition .(see [, ]) A set-valued mapW:X→Y_{is said to be} (i) strictly positive homogeneous if

W(αx) =αW(x), ∀α> ,∀x∈X;

(ii) subadditive if

W(x) +W(x)⊆W(x+x) +C.

Deﬁnition .(see [, ]) LetE⊂Xbe a nonempty convex set andG:E→Z_{be a}

set-valued map withG(x)=∅, for allx∈E.Gis said to beD-convex onE, if for anyx,x∈E

andλ∈(, ),

λG(x) + ( –λ)G(x)⊆G

λx+ ( –λ)x

+D.

LetXbe a normed space supplied with a distancedandKbe a subset ofX. We denote byd(x,K) =inf_y_∈_Kd(x,y) the distance fromxtoK, where we setd(x,∅) = +∞.

Deﬁnition .(see [, ]) Let K be a nonempty subset ofXandx∈K,u∈X. The contingent cone ofKatxis

T(K,x) :={v∈X|∃tn↓,∃vn→v, such thatx+tnvn∈K,∀n∈N}.

Proposition .(see []) Let K⊆X and x∈K.The following statements are equivalent:

(i) u∈T(K,x);

(ii) there exist sequences{λn}withλn→+∞and{xn}withxn∈Kandxn→xsuch that λn(xn–x)→v.

Proposition .(see []) Let K⊆X and x∈K.Then T(K,x)is a closed cone.

Proposition .(see []) Let K⊆X be a convex set,x∈K,and u∈T(K,x).Then TT(K,x),u=clconecone(K–x) –u.

LetEbe a nonempty subset ofX,F:E×E→Y_{be a set-valued bifunction,}_F(x

,x)=∅,

for allx,x∈E. We suppose that Y∈F(x,x), for anyx∈E.

Letx∈Ebe given.Fx:E→Yis the set-valued map deﬁned by

graph(Fx) =

(x,y)∈E×Y:y∈F(x,x)

.

The set

epi(Fx) =

(x,y)∈E×Y:y∈F(x,x) +C

is called the epigraph ofFx. Denote

Fx(E) =F(x,E) =

y∈F(x,x) :x∈E

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LetG:E→Z _{be a set-valued map with}_G(x)₌_∅_{, for all}_x_∈_E.

In this paper, we consider the set-valued vector equilibrium problem with unconstraints (USVVEP): ﬁndx∈Esuch that

F(x,x)∩(–A) =∅, for allx∈E,

whereA=A\ {Y},Ais a convex cone inY.

We also consider the set-valued vector equilibrium problem with constraints (CSVVEP): ﬁndx∈Ksuch that

F(x,x)∩(–A) =∅, for allx∈K,

whereA=A\ {Y},Ais a convex cone inY, andK:={x∈E:G(x)∩(–D)=∅}. Deﬁnition . LetintC=∅.

(i) A vectorx∈Eis called a weakly eﬃcient solution of (USTVEP) if

F(x,E)∩(–intC) =∅.

(ii) A vectorx∈Kis called a weakly eﬃcient solution of (CSTVEP) if

F(x,K)∩(–intC) =∅.

3 Second-order composed contingent epiderivatives

LetE⊂X. LetF:E→Y _{be a set-valued map,}_y

∈F(x), and (u,v)∈X×Y. We ﬁrst

recall the deﬁnition of the generalized second-order composed contingent epiderivative introduced by Liet al.[].

Deﬁnition .(see []) The generalized second-order composed contingent epideriva-tiveD_gF(x,y,u,v) ofFat (x,y) in the directive (u,v) is the set-valued map fromXtoY

deﬁned by

D_gF(x,y,u,v)(x) =MinC

y∈Y|(x,y)∈TTepi(F), (x,y)

, (u,v).

Now we introduce the following second-order composed contingent epiderivatives of set-valued maps, and then we investigate some of its properties.

Deﬁnition . Let (x,y)∈graph(F), (u,v)∈X×Y. The second-order composed

con-tingent epiderivativeDF+(x,y,u,v) ofFat (x,y) in the directive (u,v) is the set-valued

map fromXtoYdeﬁned by

DF+(x,y,u,v)(x) =

y∈Y|(x,y)∈TTepi(F), (x,y)

, (u,v), ∀x∈X.

Proposition . Let(x,ˆ ˆy)∈graph(F), (u,ˆ v)ˆ ∈X×Y,and M:=domDF+(x,ˆ y,ˆ u,ˆ ˆv).Then

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Proof Letx∈M,y∈DF+(x,ˆ y,ˆ u,ˆ v)(x)+C. Then there exist (x,ˆ y)¯ ∈T(T(epiF, (x,ˆ y)), (ˆ u,ˆ v)),ˆ

andc∈C, such that (x,y) = (x,y¯+c). Since (x,y)¯ ∈T(T(epiF, (x,ˆ y)), (ˆ u,ˆ v)), there exist se-ˆ quences (xn,yn)→(x,y) and¯ tn↓+, such that

(u,ˆ ˆv) +tn(xn,yn)∈T

epiF, (x,ˆ y)ˆ, ∀n∈N.

Moreover,∀n∈N, there exist sequences (xk

n,ykn)→(u,ˆ v) +ˆ tn(xn,yn) andtnk↓+, such that

(x,ˆ ˆy) +tk

n(xkn,ykn)∈epiF,∀k∈N. Then we have

ˆ

y+tk_nyk_n∈Fxˆ+tk_nxk_n+C, ∀n,k∈N. ()

Sincec∈C, combine with (), we haveˆy+tk

n(ykn+tnc) =yˆ+tknykn+tnktnc∈F(xˆ+tknxkn) +

C, ∀n,k∈N. That is (x,ˆ y) +ˆ tk

n(xkn,ykn+tnc)∈epiF+,∀n,k∈N. Since (xkn,ykn)→(u,ˆ v) +ˆ

tn(xn,yn), we have (xnk,ykn+tnc)→(u,ˆ v) +ˆ tn(xn,yn+c) ask→+∞. Thus,

(u,ˆ ˆv) +tn(xn,yn+c)∈T

epiF, (x,ˆ y)ˆ , ∀n∈N.

Simultaneously, (xn,yn+c)→(x,y¯+c), since (xn,yn)→(x,y) as¯ n→+∞. Together with

(x,y) = (x,¯ y¯+c), we have (x,y) =T(T(epiF, (x,ˆ y)), (ˆ u,ˆ v)), which impliesˆ

y∈DF+(x,ˆ y,ˆ u,ˆ v)(x).ˆ

So

DF+(x,ˆ y,ˆ u,ˆ v)(x) +ˆ C⊆DF+(x,ˆ y,ˆ u,ˆ v)(x).ˆ

Naturally,DF+(x,ˆ y,ˆ u,ˆ v)(x)ˆ ⊆DF+(x,ˆ y,ˆ u,ˆ ˆv)(x) +C. Thus () holds, and this completes the

proof.

By deﬁnitions and Proposition ., we can conclude that the following result holds.

Proposition . Let F:E→Y_{, (x}

,y)∈graph(F), (u,v)∈T(epi(F), (x,y)),and x∈X.

Then D_gF(x,y,u,v)(x) +C⊆DF+(x,y,u,v)(x). Remark . The inclusion relation

DF+(x,y,u,v)(x)⊆DgF(x,y,u,v)(x) +C

may not hold.

Now we give the following example to explain Remark ..

Example . LetC=R

+andF(x) ={(y,y)∈R|y≥x,y∈R},∀x∈R+. Let (x,y) =

(, (, )), and (u,v) = (, (, )). Then

TTepi(F), (x,y)

, (u,v)=x, (y,y)

∈R×R|x∈R,y≥,y∈R

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Therefore, for anyx∈R, we have

D_gF(x,y,u,v)(x) =∅

and

DF+(x,y,u,v)(x) =

(y,y)|y≥,y∈R

.

And then, for anyx∈R, we have

DF+(x,y,u,v)(x)DgF(x,y,u,v)(x) +C.

Now we discuss some crucial properties of the second-order composed contingent epi-derivative.

Proposition . Let(x,y)∈graphF, (u,v)∈T(epiF, (x,y))with v∈C and E⊂X be

convex.If F is C-convex on E,then for all x∈E,

F(x) –{y}+C⊂DF+(x,y,u,v)(x–x–u).

Proof SinceFisC-convex onE,epiFis a convex set. So it follows from Proposition . that

TTepiF, (x,y)

, (u,v)=clconeconeepiF–(x,y)

–(u,v). ()

Since for everyc∈C,x∈Sandy∈F(x), one has

(x–x–u,y–y+c)

= (x–x–u,y+c+v–y–v)∈ {x} ×

F(x) +C–(x,y)

–(u,v).

Then it follows from () that

(x–x–u,y–y)∈T

TepiF, (x,y)

, (u,v).

Thus, by the deﬁnition of the second-order composed contingent epiderivative, we have

y–y+c∈DF+(x,y,u,v)(x–x–u),

and then

F(x) –{y}+C⊂DF+(x,y,u,v)(x–x–u), ∀x∈E.

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Proposition . Let(x,y)∈graph(F), (u,v)∈T(epiF, (x,y)).Then (i) DF+(x,y,u,v)is strictly positive homogeneous.

Moreover,ifFisC-convex on a nonempty convex setE,then

(ii) DF+(x,y,u,v)is subadditive.

Proof (i) Letα>  andx∈X.

Ify∈DF+(x,y,u,v)(x), then there exist sequences{hn}withhn→+and{(xn,yn)}

with (xn,yn)∈T(epiF, (x,y)) such that

(xn,yn) – (u,v)

hn →

(x,y),

and then

(xn,yn) – (u,v) 

αhn

→(αx,αy).

So (αx,αy)∈T(T(epiF, (x,y)),u,v), and then we can obtain αy∈DF+(x,y,u,v)(αx).

Thus

αDF+(x,y,u,v)(x)⊆DF+(x,y,u,v)(x)(αx). ()

The proof of

DF+(x,y,u,v)(αx)⊆αDF+(x,y,u,v)(x)(x)

follows along the lines of (). SoDF+(x,y,u,v) is strictly positive homogeneous.

(ii) Letx,x∈X,y∈DF+(x,y,u,v)(x),y∈DF+(x,y,u,v)(x). Then one has

(x,y)∈T

TepiF, (x,y)

, (u,v), (x,y)∈T

TepiF, (x,y)

, (u,v).

SinceFisC-convex onS,epiFis convex, and thenT(T(epiF, (x,y)), (u,v)) is a close and

convex cone. Thus we have

(x+x,y+y)∈T

TepiF, (x,y)

, (u,v),

and then

y+y∈DF+(x,y,u,v)(x+x).

Thus

DF+(x,y,u,v)(x) +DF+(x,y,u,v)(x)⊂DF+(x,y,u,v)(x+x) +C,

and the proof is complete.

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Proposition . Let (x,y)∈ graph(F), (u,v) ∈ T(epiF, (x,y)), M := domDF+(x,

y,u,v).If F is C-convex on a nonempty convex set E,then DF+(x,y,u,v)(M)is a convex

cone.

4 Second-order optimality conditions of weakly efﬁcient solutions

Throughout this section, letx∈K,y= Y ∈Fx(x),intC=∅, andintD=∅. Firstly, we recall a deﬁnition and a result in [].

LetK⊂Xandx∈K. The interior tangent cone ofKatxdeﬁned as

IT(K,x) =

u∈X|∃δ>  such thatx+tu∈K,∀t∈(,δ],∀u∈BX(u,δ)

,

whereBX(u,δ) stands for the closed ball centered atu∈Xand of radiusδ. Lemma .(see []) If K⊂X is convex,x∈K,andintK=∅,then

IT(intK,x) =intcone(K–x).

Theorem . Let x be a weakly eﬃcient solution of the problem(USVVEP).Then,for

every(u,v)∈T(epiFx, (x,y))with v∈–∂C,we have

D(Fx)+(x,y,u,v)(x) +line{v}

∩(–intC) =∅, ()

for every x∈domD(Fx)+(x,y,u,v).

Proof Suppose to the contrary that there exists anx∈domD(Fx)+(x,y,u,v) such that

() does not hold. Then there existλ∈Rand

y∈D(Fx)+(x,y,u,v)(x) ()

such that

y:=y+λv∈–intC. ()

Let us consider two possible cases forλ.

Case : Ifλ> , then it follows from Proposition . and () thaty∈D(Fx)+(x,y, u,v)(x). So

x,y∈TTepiFx, (x,y)

, (u,v). ()

By deﬁnition, there exist sequences λn→+∞ and (un,vn)∈T(epiF, (x,y)) such that

(un,vn)→(u,v) and λn

(un,vn) – (u,v)

→x,y, asn→+∞. ()

It follows from (), (), andv∈–Cthat there existsN∈N such that

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Since (un,vn)∈T(epiFx, (x,y)), for everyn∈N, there exist a sequenceλkn→+∞as

k→+∞and a sequence (xk

n,ykn)∈epiF, such that (xkn,ykn)→(x,y) and λk_nxk_n,yk_n– (x,y)

→(un,vn), ask→+∞. ()

It follows from () that there existsN(n)∈N such that λk_nyk_n–y

∈–intC, ∀k>N(n),∀n>N,

which implies

yk

n–y∈–intC, ∀k>N(n),∀n>N. ()

Since (xk

n,ykn)∈epiFx, there existsy¯kn∈Fx(xkn) such thatykn∈ {¯ykn}+C. Then, by (), we

have

¯

yk_n–y∈–intC, ∀k>N(n),∀n>N.

Therefore

Fx,xkn

∩(–intC)=∅,

which contradicts that (x,y) is a weakly eﬃcient solution of the problem (USVVEP).

Case : Ifλ≤, then it follows from Propositionv∈–Cand () thaty∈–intCand (x,y)∈TTepiFx, (x,y)

, (u,v).

By a similar proof method to case , there exist consequencesxk_nandN,N(n)∈N such

that

Fx,xkn

∩(–intC)=∅, ∀k>N(n),∀n>N,

which contradicts thatxis a weakly eﬃcient solution of the problem (USVVEP). Thus

() holds, and the proof is complete.

Remark . In Theorem ., we cannot usev∈–intCinstead ofv∈–∂C. Since (x,y)

is a weakly eﬃcient solution of the problem (USVVEP),

Fx(x)∩(–intC) =∅, ∀x∈E. ()

It follows from (u,v)∈T(epi(Fx), (x,y)), andy=  that there exist sequences{λn}with λn→+∞and (xn,yn)∈epi(Fx) with (xn,yn)→(x, ) such that

λn

(xn,yn) – (x, )

→(u,v), asn→+∞.

So, it follows fromv∈–intCthat there existsN∈N such that

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which implies

yn∈–intC, ∀n>N. ()

Since (xn,yn)∈epi(Fx), there exists¯yn∈Fx(xn) such thatyn∈ {¯yn}+C. Then, combined with (), we have

¯

yn∈–intC, ∀n>N.

Therefore

Fx(xn)∩(–intC)=∅, ∀xn∈E,∀n>max{N,N}, which contradicts ().

Next, we give an example to illustrate Theorem ..

Example . LetF(x) ={(y,y)∈R|y∈R,y≥x},∀x∈R+, (x,y) = (, (, )), and

C=R

+. ThenT(epi(F), (x,y)) ={(u, (v,v))∈R×R|x∈R+,v∈R,v≥}. Take (u,v) =

(, (–, )). Then

TTepi(F), (x,y)

, (u,v)=x, (y,y)

∈R×R|x∈R,y∈R,y≥

.

Therefore, for anyx∈R, we have DF+(x,y,u,v)(x) +line{v}=

(y,y)|y∈R,y≥

.

And then, for anyx∈R, we have

DF+(x,y,u,v)(x) +line{v}

∩(–intC) =∅,

which shows that Theorem . holds.

Theorem . Let(u,v)∈T(epiFx(x,y))with v∈C and E⊂X be convex.If Fx is C-convex on E,and for all x∈E,

D(Fx)+(x,y,u,v)(x–x–u)∩(–intC) =∅, () then xis a weakly eﬃcient solution of the problem(USVVEP).

Proof It follows from Proposition . that

(Fx)(x)⊂D _(F

x)+(x,y,u,v)(x–x–u), ∀x∈E. Then, from (), we have

(Fx)(x)∩(–intC) =∅, ∀x∈E.

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Theorem . Let(u,v,w)∈T(epi(Fx,G),x,y,z)with v∈–∂C and w∈–D.If xis a weakly eﬃcient solution of(CSVVEP),then for any z∈G(x)∩(–D),

D(Fx,G)+(x,y,z,u,v,w+z)(x)

∩–coneint(C×D) +line(v,w+z)

\(Y, Z)

=∅, ()

for all x∈:=dom[D(Fx,G)+(x,y,z,u,v,w+z)].

Proof To prove the result by contradiction, suppose that there exists anx∈such that () does not hold, that is, there exists a (y,z)∈Y×Zsuch that

(y,z)∈D(Fx,G)+(x,y,z,u,v,w+z)(x) and

(y,z)∈–coneint(C×D) +line(v,w+z)

\(Y, Z)

. ()

Then, by the deﬁnition of second-order composed contingent epiderivatives, there exist sequencesλn→+∞and (un,vn,wn)∈T(epi(Fx,G), (x,y,z)) such that (un,vn,wn)→ (u,v,w+z) and

λn

(un,vn,wn) – (u,v,w+z)

→(x,y,z), asn→+∞. ()

It follows from () that there existμ> ,ν∈R,c∈intC, andd∈intDsuch that y= –μ(c+νv), z= –μd+ν(w+z)

. ()

Let us consider two possible cases forν.

Case : Ifν≤, then, from (),v∈–C, andw,z∈–D, we havey∈–intC andz∈

–intD. Thus, by (), there existsN∈N such that

λn(vn–v)∈–intC, λn

wn– (w+z)

∈–intD, ∀n>N.

Thus, it follows fromv∈–Candw∈–Dthat

vn∈–intC, wn∈–intD, ∀n>N. ()

Since (un,vn,wn)∈T(epi(Fx,G), (x,y,z)), for everyn∈N, there exist a sequence{λkn}

withλk_n→+∞ask→+∞and a sequence (xk

n,ykn,zkn)∈epi(Fx,G), such that (xkn,ykn,zkn)→

(x,y,z) and

λk_nxk_n,yk_n,zk_n– (x,y,z)

→(un,vn,wn), ask→+∞. ()

It follows from () and () that there existsN(n)∈N such that λk_nyk_n–y

∈–intC, λk_nzk_n–z

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which implies

yk_n–y∈–intC, zkn–z∈–intD, ∀k>N(n),∀n>N. ()

Since (xk

n,ykn,zkn)∈epi(Fx,G), there existy¯kn∈Fx(xkn),z¯kn∈G(xkn),c∈C andd∈Dsuch

thatyk

n=y¯kn+candznk=z¯nk+d. Then, by () andy= , we have

¯

yk

n∈–intC, ∀k>N(n), z¯kn∈–intD, ∀n>N.

So

Fx

xk_n∩–intC=∅, x_nk∈K,∀k>N(n),∀n>N,

which contradicts thatxis a weakly eﬃcient solution of (CSVVEP).

Case : Ifν> , then, from (), we gety= –μν(_νc+v) andz= –μν(_νd+ (w+z)). So it

follows fromc∈intCandd∈intDthat

y∈–intconeC+{v}, z= –intconeD+{w+z}

. ()

Then, by Lemma . and (), we gety∈IT(–intC,v) andz=IT(–intD,w+z). Therefore,

there existsδ>  such that

v+δy∈–intC, ∀y∈BY(y,δ), ()

w+z+δz∈–intD, ∀y∈BZ(z,δ). ()

For thisδ, it follows from () that there existsN∈N such that

δλn> , λn(vn–v)∈BY(y,δ), λn

wn– (w+z)

∈BZ(z,δ), ∀n>N.

Then, by () and (), we have

vn–  –



δλn

v∈–intC, wn–  –



δλn

(w+z)∈–intD, ∀n>N.

Thus, fromv∈–C,w,z∈–D, andδλn> ,∀n>N, we have

vn∈–intC, wn∈–intD, ∀n>N.

By a similar proof method to case , there existsN(n)∈N such that

Fx

xk_n∩–intC=∅, x_nk∈K,∀k>N(n),∀n>N,

which contradicts thatxis a weakly eﬃcient solution of (CSVVEP). Thus () holds and

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Theorem . Let(u,v,w)∈T(epi(Fx,G),x,y,z)with v∈–∂C and w∈–D.If xis a weakly eﬃcient solution of(CSVVEP),then for any z∈G(x)∩(–D),

D(Fx,G)+(x,y,z,u,v,w+z)(x)∩–

int(C×D) +(,z)

=∅, ()

for all x∈:=dom[D(Fx,G)+(x,y,z,u,v,w+z)].

Proof To prove the result by contradiction, suppose that there exists anx∈such that () does not hold, that is, there exists (y,z)∈Y×Zsuch that

(y,z)∈D(Fx,G)+(x,y,z,u,v,w+z)(x), (y,z)∈–int(C×D) +(,z)

. ()

Then, by the deﬁnition of second-order composed contingent epiderivatives, there exist sequencesλn→+∞and (un,vn,wn)∈T(epi(Fx,G), (x,y,z)) such that (un,vn,wn)→ (u,v,w+z) and

λn

(un,vn,wn) – (u,v,w+z)

→(x,y,z), asn→+∞. ()

It follows from () that there existc∈intCandd∈intDsuch that

y= –c, z= –(d+z). ()

Thus, by (), there existsN∈N such that

λn> , λn(vn–v)∈–intC, λn

wn– (w+z)

∈–(intD+z), ∀n>N.

Thus, it follows fromv∈–Candw,z∈–Dthat

vn∈–intC, wn∈–intD, ∀n>N. ()

Since (un,vn,wn)∈T(epi(Fx,G), (x,y,z)), for everyn∈N, there exist a sequence{λkn}

withλk_n→+∞ask→+∞and a sequence (xk

n,ykn,znk)∈epi(Fx,G), such that (x

k

n,ykn,zkn)→

(x,y,z) and

λk_nxk_n,yk_n,zk_n– (x,y,z)

→(un,vn,wn), ask→+∞. ()

It follows from () and () that there existsN(n)∈N such that

λk_nyk_n–y

∈–intC, λk_nzk_n–z

∈–intD, ∀k>N(n),∀n>N,

which implies

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Since (xk

n,ykn,zkn)∈epi(Fx,G), there existy¯kn∈Fx(xkn),z¯kn∈G(xkn),c∈C, andd∈Dsuch

thatyk

n=y¯kn+candznk=z¯nk+d. Then, by () andy= , we have

¯

y_nk∈–intC, ∀k>N(n), z¯kn∈–intD, ∀n>N.

So

Fx

xk_n∩–intC=∅, x_nk∈K,∀k>N(n),∀n>N,

which contradicts thatxis a weakly eﬃcient solution of (CSVVEP). Thus () holds and

the proof is complete.

Theorem . Let E⊂X be a nonempty convex set,z∈G(x)∩(–D)and(u,v,w)∈X×

(–C)×(–D).Suppose that the following conditions are satisﬁed:

(i) (Fx,G)isC×D-convex onE.

(ii) xis a weakly eﬃcient solution of(CSVVEP).

Then there existφ∈C∗andψ∈D∗,not both zero functionals,such that

inf

(y,z)∈A

φ(y) +ψ(z)

=  and ψ(z) = ,

where A:=_x_∈D(Fx,G)+(x,y,z,u,v,w+z)(x)and:=dom[D(Fx,G)+(x,y,z, u,v,w+z)].

Proof DeﬁneM=A+ (Y,z). By Proposition ., we see thatMis a convex set. By

The-orem ., we get

M∩–int(C×D)=∅.

By the separation theorem of convex sets, there existφ∈Y∗andψ ∈Z∗, not both zero functionals, such that

φ(y) +ψ(z)≥φ(¯y) +ψ(¯z), for all (y,z)∈M, (¯y,z)¯ ∈–int(C×D). ()

SinceintC∪ {Y}andintD∪ {Z}are cones, by (), we have

φ(y)¯ ≤ψ(z),¯ for all (y,¯ ¯z)∈–intC×intD, ()

and

φ(y) +ψ(z)≥, for all (y,z)∈M. ()

From (), we ﬁnd thatψ is bounded below onintD. Thenψ(z)≥, for allz∈intD. Naturallyψ∈D∗.

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It follows from Proposition . that (Y, Z)∈A, and then, fromz∈–D,ψ∈D∗, and

(), we obtain

inf

(y,z)∈A

φ(y) +ψ(z)

=  and ψ(z) = .

The proof is complete.

Theorem . Let E⊂X be a nonempty convex set, (u,v,w)∈T(epi(Fx,G),x,y,z)with v∈C and w∈D and z∈G(x)∩(–D).Suppose that the following conditions are satisﬁed:

(i) (Fx,G)isC×D-convex onE;

(ii) there existφ∈C∗\ {}andψ∈D∗such that

inf

(y,z)∈V

φ(y) +ψ(z)

=  and ψ(z) = ,

where U :=_x_∈D(Fx,G)+(x,y,z,u,v,w)(x) and V :=dom[D(Fx,G)+(x,y,z,u, v,w)].

Then xis a weakly eﬃcient solution of(CSVVEP).

Proof To prove the result by contradiction, suppose thatxis not a weakly eﬃcient

solu-tion of (CSVVEP). Then there existx∈Kandy∈F(x,x) such thaty∈–intC. Since

x∈K, there existsz∈G(x)∩(–D). It follows from assumption (i) and Proposition . that we have

y–y,z–z

∈U,

and then, from assumption (ii), we obtain

φy–y

+ψz–z

≥. ()

Sincey–y=y∈–intC,φ∈C∗\ {},φ(y–y) < . It follows fromz∈G(x)∩(–D), ψ∈D∗, andψ(z) =  thatψ(z–z)≤, thus

φy–y

+ψz–z

< ,

which contradicts (). Soxis a weakly eﬃcient solution of (CSVVEP), and this

com-pletes the proof.

5 Conclusions

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Author details

1_{College of Sciences, Chongqing Jiaotong University, Chongqing, 400074, China.}2_{College of Mathematics and Statistics,}

Chongqing Technology and Business University, Chongqing, 400067, China.

Acknowledgements

The authors would like to thank anonymous referees for their valuable comments and suggestions, which helped to improve the paper. This research was partially supported by Science and Technology Research Project of Chongqing Municipal Education Commission (KJ130414), Chongqing Natural Science Foundation Project of CQ

CSTC(cstc2012jjA00038) and the National Natural Science Foundation of China (Nos: 11171362, 11271389, 11201509, and 11301571).

Received: 15 February 2014 Accepted: 29 September 2014 Published:16 Oct 2014

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10.1186/1029-242X-2014-406