Coderivatives of gap function for Minty vector variational inequality

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R E S E A R C H

Open Access

Coderivatives of gap function for Minty

vector variational inequality

Xiaowei Xue

1*

_{and Yu Zhang}

2

*_{Correspondence:} [email protected] 1_{College of Mathematics and} Statistics, Nanyang Normal University, Nanyang, Henan 473061, China

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

Abstract

The purpose of this paper is to investigate coderivatives of the gap function involving the Minty vector variational inequality. First, we discuss the regular coderivative, the normal coderivative, and the mixed coderivative of a class of set-valued maps. Then, by using the relationships between the coderivatives of a set-valued map and its eﬃcient points set-valued map, we obtain the coderivatives of the gap function for the Minty vector variational inequality.

MSC: 49K40; 90C29; 90C31; 49J53

Keywords: Minty vector variational inequality; gap function; normal cone; coderivative

1 Introduction

The vector variational inequality (for short, VVI) and the Minty vector variational inequal-ity (for short, MVVI) have been of great interest in the academic and professional com-munities ever since the path-breaking paper [] in the early s. Enormous results on the existence (see [, ]) and stabilities (see [, ]) have been obtained. There are some applications to be found in vector traﬃc equilibrium problems (see [, ]).

It is well known that the concept of gap functions is very important for the study of (VVI) and (MVVI). From the vector optimization point of view, Chenet al.[] defined the gap function for the (VVI) problem as a set-valued map. Under some suitable coerciveness conditions, Liet al.[] discussed the differential and sensitivity properties of the set-valued gap functions defined in [] for (VVI). They also obtained an explicit expression of the contingent derivative for a class of set-valued maps, and some optimality conditions for (VVI) and weak (VVI) by virtue of the gap functions. Later, by the definition of the gap function for Minty vector variational inequalities, some differential and sensitivity results for Minty vector variational inequalities were also obtained in []. High-order optimality conditions and differential and sensitivity properties for gap functions of weak (VVI) were also considered (see []).

The generalized derivatives mentioned above for set-valued maps are generated by tan-gent cones to their graphs in primal spaces. Another derivative-like construction for set-valued maps has been introduced by Mordukhovich [], which is called coderivatives and is generated by normal cones to their graphs in dual spaces. There are numerous appli-cations of coderivatives and the corresponding subdiﬀerential to derive necessary

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tions and existence properties in various vector optimization problems, such as [–]. Coderivatives have also been applied to sensitivity analysis of scalar (single-objective) op-timization problems. We refer the readers to [–] for just a few of them.

Recently, Li and Xue [] discussed the differential and sensitivity properties of the set-valued gap functions defined in [] for (VVI) via coderivatives. First, they established an explicit expression for computing the normal coderivative and mixed coderivative of a class of set-valued map. Then, through discussing the relations between a set-valued map and its efficient points set-valued map, they investigated sensitivity properties of the gap function for VVI. They also obtained some optimality conditions for (VVI).

Motivated by the work reported in [, ], in this paper, we make an eﬀort to investigate the coderivatives of Minty vector variational inequality problem in general Banach spaces. First, we establish an explicit expression for computing the regular coderivatives, normal coderivative, and mixed coderivative of a class of set-valued maps. Then, using the rela-tions between coderivatives of a set-valued map and its eﬃcient points set-valued map, we obtain the coderivatives of the gap function for the Minty vector variational inequality. We also give some examples to illustrate the results.

The rest of the paper is organized as follows. In Section , we recall the basic defini-tions and notadefini-tions from the vector variational inequality, set-valued analysis, and vari-ational analysis. In Section , we establish the coderivative of a class of set-valued map. Under some mild conditions, we first give the including relations of the coderivatives of set-valued maps. Then we obtain the explicit expressions under some stronger conditions. In Section , we give the coderivatives of the gap function for (MVVI).

Throughout the paper we use the standard notation, with special symbols introduced where they are deﬁned. Unless otherwise stated, all spaces considered are Banach spaces, whose norms are always denoted by · . For any spaceX, we consider its dual spaceX∗

equipped with the weak* topologyw∗, where·,·means the canonical pairing. The closed ball with centerxand radiusηis denoted byBη(x). The symbolA∗ denotes the adjoint operator of a linear continuous operatorA. IfF:X⇒Y is a set-valued map, we denote bydomF={x∈X|F(x)=∅}andgphF={(x,y)∈X×Y|y∈F(x)}, the domain and graph ofF, respectively. The notation→∗ stands for weak∗ convergence in a dual space, while

xn→S xmeans that the sequencexnis contained in the subsetSand converges tox. For the set-valued mapF:X⇒X∗the expression

Limsup

x→¯x

F(x) =x∗∈X∗| ∃sequencesxk→ ¯x,x_k∗→∗ x∗s.t.x∗_k∈F(xk) for allk∈N

signiﬁes the sequential Painlevé-Kuratowski upper (outer) limit with respect to the norm topology inXand the weak* topology in X∗. The origins of all real normed spaces are denoted by .

2 Basic deﬁnitions and preliminaries

Throughout this paper, letL(X,Y) be the set of all linear continuous operators fromX

toY. For anyA∈L(X,Y), we introduce norm

AL=supA(x)| x ≤

.

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Deﬁnition . LetF:X→L(X,Y) be a vector-valued function.Fis said to be Fréchet diﬀerentiable atxif and only if there exists a linear continuous operator:X→L(X,Y), such that

lim

x→x

F(x) –F(x) –(x–x)L x–x = .

Obviously,is unique. We denote derivativeofF atxby ∇F(x). If, for any x∈ K,Fis Fréchet diﬀerentiable atx,Fis said to be Fréchet diﬀerentiable onK. Therefore,

∇F(·) :X→L(X,Y) is a vector-valued function.

In the following of this section, we introduce the basic concepts and constructions of variational analysis and generalized diﬀerentiation needed for formulations and justiﬁca-tions of the main results of the paper. Most of the concepts and properties can be found in [].

Deﬁnition .[, ] Let⊂Xbe a nonempty subset of a Banach space. (i) Givenx¯∈andε≥. The set ofε-normalstoatx¯∈is deﬁned by

ˆ

Nε(x¯,) =

x∗∈X∗lim sup

x→¯x

x∗,x–x¯ x–x¯ ≤ε

. ()

Whenε= , the set () is a cone that is called the regular normal cone (or the

prenormal cone) toatx¯and is denoted byNˆ(x¯,). We putNˆε(x¯,) =∅for all

ε≥ifx¯∈/.

(ii) The Mordukhovich normal cone (or basic normal cone) to⊂Xatx¯is deﬁned through the Painlevé-Kuratowski upper (outer) limit as

N(x¯,) = Limsup

xk→¯x,εk→+

ˆ

Nε_k(xk,). ()

Deﬁnition .[, ] Consider a set-valued map:X⇒Ybetween Banach spaces. (i) Theε-coderivativeDˆ∗ε(x¯,y¯)at(x¯,y¯)is deﬁned through theε-normal set () to the

graph as

ˆ

D∗_ε(x¯,y¯)y∗ =x∗∈X∗|x∗, –y∗ ∈ ˆNε

(x¯,y¯),gph . ()

Whenε= , the positive homogeneous set-valued map ofy∗in () is called the regular coderivative ofat(x¯,y¯)and denoted byDˆ∗(x¯,y¯)(·).

(ii) The normal (Mordukhovich) coderivative ofat(x¯,¯y)is

D∗_N(x¯,y¯)y∗ =x∗∈X∗|x∗, –y∗ ∈N(x¯,y¯),gph , ()

that is,D∗_N(x¯,y¯)(y∗)is the collection of allx∗for which there are sequences

εk→+,(xk,yk)→(x¯,¯y),(x∗k,y∗k)

∗

→(x∗,y∗)with(xk,yk)∈gphand

x∗_k∈ ˆD∗εk(xk,yk)(y ∗

k).

(iii) The mixed coderivativeD∗_M(x¯,y¯)of a set-valued map:X⇒Yat(x¯,y¯)is the set-valued mapD∗_M(x¯,y¯) :Y∗⇒X∗deﬁned by

D∗_M(x¯,y¯)y∗ = Limsup

(xk,yk,y∗k)→(x,¯y,y¯ ∗),εk→+

ˆ

D∗εk(xk,yk)

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i.e.,x∗∈D∗_M(x¯,y¯)(y∗)if and only if there are sequencesεk→+,

(xk,yk,y∗_k)→(x¯,y¯,y∗),x∗_k→∗ x∗with(xk,yk)∈gph, andx_k∗∈ ˆD∗εk(xk,yk)(y ∗

k).

It follows from the deﬁnitions thatD∗_M(x¯,y¯)(y∗)⊂D∗_N(x¯,y¯)(y∗) when the equality ob-viously holds if Y is ﬁnite-dimensional. We say that is regular atx¯∈ if N(x¯,) =

ˆ

N(x¯,) andisN-regular (resp.M-regular) at (x¯,y¯) if and only ifD∗_N(x¯,¯y) =Dˆ∗(x¯,¯y) (resp.D∗_M(x¯,y¯) =Dˆ∗(x¯,y¯)) (see []). The following proposition gives a suﬃcient con-dition for the regularity ofand special representations of the coderivatives.

Proposition .[] Let:X→Y be Fréchet diﬀerentiable atx¯.Then

ˆ

D∗(x¯)y∗ =∇(x¯) ∗y∗, ∀y∗∈Y∗.

Moreover,ifis strictly diﬀerentiable atx¯,i.e.,is single-valued aroundx and¯

lim

x,x→¯x

(x) –x –∇(x¯)x–x /x–x= ,

thenis N-regular atx and one has¯

D∗_N(x¯)y∗ =D∗_M(x¯)y∗ =∇(x¯) ∗y∗, ∀y∗∈Y∗.

We also need some Lipschitzian notions in the following study.

Deﬁnition .[] Letf :X→Y be a single-valued map andx¯∈domf.f is said to be local upper Lipschitzian atx¯if there are numbersη>  andL>  such that

f(x) –f(x¯)≤Lx–x¯, for allx∈Bη(x¯)∩domf.

We say that a set-valued mapF:X⇒Y admits a local upper Lipschitzian selection at (x¯,y¯)∈gphFif there is a single-valued mapf :domF→Y which is local upper Lips-chitzian atx¯satisfyingf(x¯) =y¯andf(x)∈F(x) for allx∈domFin a neighborhood ofx¯.

Deﬁnition .[] We say that the domination property holds for the multifunctionF:

X⇒Yaroundx¯if there exists a neighborhoodUofx¯such that

F(x)⊂MinF(x) +S, for allx∈U,

or

F(x)⊂MaxF(x) –S, for allx∈U.

3 Coderivatives of a set-valued map

In subsequent sections, letK be a closed subset ofX,F:X→L(X,Y) be a continuous vector-valued map, and

G(x) =

z∈K

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We will discuss the coderivative properties of G. First, recall that a set-valued map H:

X⇒Y is said to be inner semicontinuous at (xˆ,yˆ)∈gphHif for every sequencexk→ ˆx

with xk∈domH there is a sequenceyk∈H(xk) converging toyˆ ask→ ∞. H is inner

semicompact atxˆif for every sequencexk→ ˆxwithxk∈domHthere is a sequenceyk∈ H(xk) that contains a convergent subsequence ask→ ∞.

In the rest of this paper, let D∗ stand either for the normal coderivative () or for the mixed coderivative (). Since the proof methods of normal coderivative and mixed coderivative are similar, we only show the case of normal coderivative in the following.

Theorem . Letxˆ∈domG,yˆ∈G(xˆ),and

M(x,y) =z∈K|F(z)(x–z) =y. ()

(i) For anyy∗∈Y∗,

ˆ

D∗G(xˆ,yˆ)y∗ ⊂

ˆ

z∈M(x,ˆy)ˆ

F(zˆ)∗y∗+Nˆ(xˆ,domG). ()

(ii) IfMis inner semicompact at(xˆ,yˆ),then,for anyy∗∈Y∗,

ˆ

z∈M(x,ˆy)ˆ

F(zˆ)∗y∗+N(xˆ,domG). ()

(iii) Givenzˆ∈M(xˆ,ˆy),ifMis inner semicontinuous at(xˆ,yˆ,zˆ),then,for anyy∗∈Y∗,

D∗G(xˆ,yˆ)y∗ ⊂F(zˆ)∗y∗+N(xˆ,domG). ()

Proof (i) For anyx∗∈ ˆD∗G(xˆ,yˆ)(y∗), by the deﬁnitions of regular coderivative and regular normal cone,

lim sup

(xk,yk)

gphG −→(x,ˆˆy)

(x∗, –y∗), (xk–xˆ,yk–yˆ)

xk–xˆ+yk–yˆ ≤.

Then, for anyzk∈Ksatisfyingzk→ ˆz, we have

lim sup

xk

domG −→ ˆx

x∗,xk–xˆ–y∗,F(zk)(xk–zk) –F(zˆ)(xˆ–zˆ)

xk–xˆ+F(zk)(xk–zk) –F(zˆ)(xˆ–zˆ) ≤.

Especially, letzk=zˆ, we have

lim sup

xk

domG −→ ˆx

x∗,xk–xˆ–y∗,F(ˆz)(xk–xˆ) xk–xˆ+F(zˆ)(xk–xˆ) ≤.

Thus,

lim sup

xk

domG −→ ˆx

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This ensures thatx∗–F(zˆ)∗y∗∈ ˆN(xˆ,domG). So we have

ˆ

D∗G(xˆ,yˆ)y∗ ⊂F(zˆ)∗y∗+Nˆ(xˆ,domG).

(ii) For anyx∗∈D∗G(xˆ,yˆ)(y∗), by the deﬁnitions of coderivatives and normal cone, there are sequences of vector (xk,yk)→(xˆ,yˆ), (x∗_k,y∗_k)→∗ (x∗,y∗) (y∗_k→y∗for the case of a mixed coderivative) andεk↓ such thatyk∈G(xk) and

lim sup

(x_ki,y_ki)gph−→G(xk,yk)

(x∗_k, –y∗_k), (xki–xk,yki–yk)

xki–xk+yki–yk

≤εk.

SinceMis inner semicompact at (xˆ,ˆy), for the above (xk,yk), there exists a sequence of

zk∈M(xk,yk) that contains a subsequence converging to someˆz. SinceKis closed, we have

ˆ

z∈K. Letk→ ∞inyk=F(zk)(xk–zk), we haveˆy=F(ˆz)(xˆ–zˆ), which implieszˆ∈M(xˆ,yˆ).

Then, for anyzki∈Ksatisfyingzki→zk, we have

lim sup

x_kidom−→Gxk

x∗_k,xki–xk–y ∗

k,F(zki)(xki–zki) –F(zk)(xk–zk)

xki–xk+F(zki)(xki–zki) –F(zk)(xk–zk)

≤εk.

Especially, letzki=zk, we have

lim sup

x∗_k,xki–xk–y∗k,F(zk)(xki–xk)

xki–xk+F(zk)(xki–xk)

≤εk.

Thus,

lim sup

x∗_k,xki–xk–F(zk)∗y∗k, (xki–xk)

xki–xk

≤ε_k,

whereε_k= ( +F(zk))εk→ ask→ ∞. This ensures thatx∗–F(zˆ)∗y∗∈N(xˆ,domG).

So we have

ˆ

z∈M(x,ˆy)ˆ

F(zˆ)∗y∗+N(xˆ,domG).

(iii) It can be proved similarly to the case (ii), since for any sequence (xk,yk)→(xˆ,yˆ), by the inner semicontinuous assumption ofM, there exists a sequencezk∈M(xk,yk)

con-verging tozˆ. This complete the proof.

Now we turn to the converse inclusion.

Theorem . Letxˆ∈domG,yˆ∈G(xˆ),andzˆ∈M(xˆ,yˆ).If F is local upper Lipschitzian atˆz,and M admits a local upper Lipschitzian selection at(xˆ,yˆ,zˆ),then,for any y∗∈Y∗ satisfying F(zˆ)∗y∗∈ ˆD∗F(zˆ)(y∗(xˆ–zˆ)∗),

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Moreover,ifdomG is regular atxˆ,then,for any y∗∈Y∗satisfying F(zˆ)∗y∗∈ ˆD∗F(ˆz)(y∗(xˆ–

ˆ z)∗),

F(zˆ)∗y∗+N(xˆ,domG)⊂D∗G(xˆ,yˆ)y∗ .

Proof LetF(zˆ)∗y∗∈ ˆD∗F(ˆz)(y∗(xˆ–ˆz)∗) andx∗∈ ˆN(xˆ,domG). Then we have

lim sup

z→ˆFz

F(zˆ)∗y∗,z–zˆ–y∗(xˆ–zˆ)∗,F(z) –F(ˆz)

z–ˆz+F(z) –F(zˆ)L ≤



and

lim sup

xdom→ ˆGx

x∗,x–xˆ x–xˆ ≤.

This means that, for anyε> , one can ﬁndη> ,η>  such that

y∗,F(zˆ)(z–ˆz)–y∗,F(z) –F(zˆ) (xˆ–zˆ)≤εz–zˆ+F(z) –F(ˆz)_L

and

x∗,x–xˆ≤εx–xˆ,

for allz∈Bη(zˆ),x∈Bη(xˆ).

SinceMadmits a local upper Lipschitzian selection at (xˆ,yˆ,zˆ), for any (x,y) gphG −→(xˆ,yˆ), there are a constantt>  andz∈M(x,y) such thatz–ˆz ≤t(x–xˆ+y–yˆ). Further-more, the locally upper Lipschitzian assumption ofFensures thatF(z) –F(zˆ)L≤Lz–zˆ

for someL>  wheneverz→ ˆz. Thus, for any (x,y) gphG

−→(xˆ,ˆy), we have the inequalities

x∗+F(zˆ)∗y∗,x–xˆ–y∗,y–ˆy

=x∗,x–xˆ+F(zˆ)∗y∗,x–xˆ–y∗,F(z)(x–z) –F(zˆ)(xˆ–zˆ)

=x∗,x–xˆ+y∗,F(zˆ)(z–zˆ)–y∗,F(z) –F(ˆz) (xˆ–zˆ)

–y∗,F(z) –F(zˆ) (x–xˆ) – (z–zˆ)

≤εx–xˆ +εz–zˆ+F(z) –F(zˆ) –y∗,F(z) –F(zˆ) (x–xˆ) – (z–zˆ)

≤εx–xˆ +εz–zˆ+Lz–zˆ +Lz–zˆy∗(x–xˆ) – (z–ˆz)

≤(Lt+t+ )εx–xˆ+y–yˆ +L(t+ )z–zˆy∗x–xˆ+y–yˆ

=(Lt+t+ )ε+Lt(t+ )y∗x–xˆ+y–yˆ x–xˆ+y–ˆy .

Sinceε>  is chosen arbitrarily, we have

lim sup

(x,y)gph−→G(x,ˆy)ˆ

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By the deﬁnition of regular coderivative,x∗+F(zˆ)∗y∗∈ ˆD∗G(xˆ,yˆ)(y∗), and then

F(zˆ)∗y∗+Nˆ(xˆ,domG)⊂ ˆD∗G(xˆ,yˆ)y∗ .

Moreover, if K is regular atxˆ, thenN(xˆ,domG) =Nˆ(xˆ,domG) and the residual part is

obvious.

Remark . In fact, since the local upper Lipschitzian selection ofMimplies the inner semicontinuity and inner semicompactness ofM, the converse include relations in Theo-rem . can be written as equalities.

Corollary . Letzˆ∈M(xˆ,ˆy).If F is Fréchet diﬀerentiable atˆz,and M admits a local upper Lipschitzian selection at(xˆ,yˆ,zˆ),then,for any y∗∈Y∗ satisfying F(ˆz)∗y∗=∇F(zˆ)∗(y∗(xˆ–

ˆ

z)∗),we have

ˆ

D∗G(xˆ,yˆ)y∗ =F(zˆ)∗y∗+Nˆ(xˆ,domG).

Moreover,ifdomG is regular atxˆ,then G is N-regular at(xˆ,yˆ)and for any y∗∈Y∗satisfying F(zˆ)∗y∗=∇F(ˆz)∗(y∗(xˆ–ˆz)∗),we have

D∗G(xˆ,yˆ)y∗ =F(zˆ)∗y∗+N(xˆ,domG).

Proof SinceFis Fréchet diﬀerentiable atzˆ, thenFis locally upper Lipschitzian atxˆ, and for anyy∗∈Y∗,Dˆ∗F(zˆ)(y∗(xˆ–zˆ)∗) =∇F(zˆ)∗(y∗(xˆ–zˆ)∗). The ﬁrst equality relation immediately follows from Theorems ., ..

Now assume thatdomGis regular atxˆ, we prove the second part of the corollary. Obvi-ously, the upper Lipschitz selection property ofMimplies thatMis inner semicontinuous at (xˆ,ˆy,zˆ). Thus, for anyy∗∈Y∗satisfyingF(zˆ)∗y∗=∇F(ˆz)∗(y∗(xˆ–ˆz)∗), we have

D∗G(xˆ,yˆ)y∗ ⊂F(zˆ)∗y∗+N(xˆ,domG) =F(zˆ)∗y∗+Nˆ(xˆ,domG)

⊂ ˆD∗G(xˆ,yˆ)y∗ ⊂D∗G(xˆ,yˆ)y∗ .

The proof is completed.

We give an example to illustrate Theorems . and ..

Example . LetX=Y=R,K= [–, ],S=R+, andF(x) =x. Letxˆ= –,yˆ= . Then we

have

G(x) = ⎧ ⎪ ⎨ ⎪ ⎩

[,x+ ], x> ; [x– ,x+ ], –≤x≤; [x– , ], x≤–.

By direct computing, we haveM(xˆ,yˆ) ={–, },Nˆ(xˆ,domG) =N(xˆ,domG) ={},

ˆ

D∗G(xˆ,yˆ)y∗ =

, y∗= ;

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and

D∗G(xˆ,yˆ)y∗ =

{y∗, }, y∗≤;

∅, else.

Then, for anyy∗∈R, we have

ˆ

z∈M(x,ˆy)ˆ

F(zˆ)∗y∗+Nˆ(xˆ,domG)

and

ˆ

z∈M(x,ˆy)ˆ

F(zˆ)∗y∗+N(xˆ,domG).

However, for anyzˆ∈M(xˆ,yˆ), the inclusion

D∗G(xˆ,yˆ)y∗ ⊂F(zˆ)∗y∗+N(xˆ,domG)

does not hold. This is so becauseMis not inner semicontinuous at any (xˆ,yˆ,zˆ). For ex-ample, let (xˆ,ˆy,zˆ) = (–, , –), we can ﬁnd a sequence (xk,yk) = (– –_k, ) which does not

have a sequencezk∈M(xk,yk) converging tozˆask→ ∞.

On the other hand, since Mdoes not admit a local upper Lipschitzian selection at (xˆ,ˆy,zˆ) = (–, , –), the converse inclusions

F(zˆ)∗y∗+Nˆ(xˆ,domG)⊂ ˆD∗G(xˆ,yˆ)y∗

and

F(zˆ)∗y∗+N(xˆ,domG)⊂D∗G(xˆ,yˆ)y∗

do not hold for anyy∗∈RthoughFis continuous diﬀerential atˆzanddomGis regular atxˆ.

4 Coderivatives of gap functions

LetX,Ybe Banach spaces andS⊂Ybe a closed convex and point cone. Given a nonempty closed setK⊂Xand a mapF:K→L(X,Y), whereL(X,Y) is a set of all linear continuous operators fromXtoY, the Minty vector variational inequality is to ﬁndx∗∈Ksuch that

F(x)x–x∗ ∈/–S\ {Y}, ∀x∈K.

Deﬁnition . LetSbe a closed convex and pointed cone inY with nonempty interior.

A set-valued mapN:X⇒Yis said to be a gap function of (MVVI) if and only if (a) ∈N(xˆ)if and only ifxˆsolves (MVVI);

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Consider set-valued mapN:X⇒Y deﬁned by

N(x) :=Max_SG(x) =Max_S

z∈K

F(z)(x–z), x∈K,

where the symbolMaxSdenotes the collection of eﬃcient points. That is, for a setA∈Y,

MaxSA:=

a∈A|a∈As.t.a–a∈S\ {Y}

.

By Theorem . in [],Nis a gap function of (MVVI). So, (MVVI) is equivalent to the following set-valued optimization problem:

MinSN(x) subject to x∈K.

In this section, we discuss the coderivativeD∗N.

Theorem . Let K be a compact set,xˆ∈K,yˆ∈N(xˆ),and

M(x,y) =z∈K|F(z)(x–z) =y.

(i) For anyy∗∈Y∗satisfyingsup_s_∈_S_\{__}y∗_s,s=:v< ,

ˆ

D∗N(xˆ,yˆ)y∗ ⊂

ˆ

z∈M(x,ˆˆy)

F(zˆ)∗y∗+Nˆ(xˆ,K). ()

(ii) IfMis inner semicompact at(xˆ,yˆ),then,for anyy∗∈Y∗satisfying

sup_s_∈_S_\{__}y∗,s

s =:v< ,

D∗_MN(xˆ,ˆy)y∗ ⊂

ˆ

z∈M(x,ˆy)ˆ

F(ˆz)∗y∗+N(xˆ,K). ()

(iii) Givenzˆ∈M(xˆ,ˆy),ifMis inner semicontinuous at(xˆ,yˆ,zˆ),then,for anyy∗∈Y∗ satisfyingsup_s_∈_S_\{__}y∗_s,s=:v< ,

D∗_MN(xˆ,ˆy)y∗ ⊂F(zˆ)∗y∗+N(xˆ,K). ()

In addition to the assumption of case(ii)and(iii),respectively,if K has a compact base,

then the including relations in case(ii)and(iii)hold for normal coderivatives.

Proof (i) First, we show thatGis compact at anyx∈K, which ensures thatGis locally compact aroundxˆ. Give a sequence{(xi,yi)} ⊂gphGsatisfyingxi→x. By the construction

ofG, there existzi∈K such thatyi=F(zi)(x–zi). SinceK is a compact set, we assume without loss of generality thatzi→z∈K. The continuity ofFimplies thatyi=F(zi)(x–

zi)→F(z)(x–z) :=y∈G(x). Therefore,Gis compact at any x∈K and then G(x) is a compact set for anyx∈K.

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we have –G(x) +S=MinS(–G(x)) +S, which implies –G(x) +S= –N(x) +S. Thus, for any y∗∈Y∗, by the deﬁnitions of regular coderivatives, we have

ˆ

D∗(–G+S)(xˆ, –yˆ)–y∗ =Dˆ∗(–N+S)(xˆ, –yˆ)–y∗ .

Obviously –Gis also compact at any x∈K. According to [], Proposition ., this implies that –Nis locally order semicontinuous around (xˆ, –yˆ) and then –Gis order semi-continuous at (xˆ, –yˆ). Employing [], Proposition ., we have, for anyy∗∈Y∗satisfying sup_s_∈_S_\{__}y∗_s,s=:v< ,

ˆ

D∗N(xˆ,yˆ)y∗ =Dˆ∗(–N)(xˆ, –yˆ)–y∗

=Dˆ∗(–N+S)(xˆ, –yˆ)–y∗

=Dˆ∗(–G+S)(xˆ, –ˆy)–y∗

=Dˆ∗(–G)(xˆ, –ˆy)–y∗

=Dˆ∗G(xˆ,yˆ)y∗ .

Note thatdomG=K, the inclusions follow from Theorem ..

(ii) Using [], Proposition ., we get, for anyy∗∈Y∗satisfyingsup_s_∈_S_\{__}y∗,s

s =:v< ,

D∗_M(–N)(xˆ, –yˆ)–y∗ =D∗_M(–N+S)(xˆ, –ˆy)–y∗

and

D∗_M(–G+S)(xˆ, –yˆ)–y∗ ⊂D∗_M(–G)(xˆ, –yˆ)–y∗ .

Thus, case (ii) and (iii) can be proved similar to case (i).

Moreover, if forKwe have the compact case, then according to [], Proposition ., for anyy∗∈Y∗satisfyingsup_s_∈_S_\{__}y∗_s,s=:v< , we have

D∗_N(–N)(xˆ, –ˆy)–y∗ =D∗_N(–N+S)(xˆ, –yˆ)–y∗

and

D∗_N(–G+S)(xˆ, –yˆ)–y∗ ⊂D∗_N(–G)(xˆ, –yˆ)–y∗ .

Thus, the including relations in case (ii) and (iii) hold for normal coderivatives. This

com-pletes the proof.

Example . LetX,Y,K,F, (xˆ,yˆ) as in Example .. Obviously

N(x) =

x+ , x∈K;

∅, else.

By direct computing, we haveNˆ(xˆ,K) =N(xˆ,K) =R–and

ˆ

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Then, for anyy∗∈Y∗satisfyingsup_s_∈_S_\{__}y∗_s,s=:v< ,i.e.,y∗< ,

ˆ

z∈M(x,ˆˆy)

F(zˆ)∗y∗+Nˆ(xˆ,K)

and

ˆ

z∈M(x,ˆˆy)

F(zˆ)∗y∗+N(xˆ,K).

Theorem . Let K be a compact set,xˆ∈K,yˆ∈N(xˆ),andzˆ∈M(xˆ,yˆ).

(i) IfFis local upper Lipschitzian relative toKatzˆ,andMadmits a local upper Lipschitzian selection at(xˆ,ˆy,zˆ),then,for anyy∗∈Y∗satisfying

F(zˆ)∗y∗∈ ˆD∗F(zˆ)(y∗(xˆ–zˆ)∗)andsup_s_∈_S_\{__}y∗,s

s =:v< ,

F(zˆ)∗y∗+Nˆ(xˆ,K)⊂ ˆD∗N(xˆ,ˆy)y∗ .

(ii) In addition to the conditions in(i),ifKis regular atxˆ,then,for anyy∗∈Y∗satisfying F(zˆ)∗y∗∈ ˆD∗F(zˆ)(y∗(xˆ–zˆ)∗)andsup_s_∈_S_\{__}y∗,s

s =:v< ,

F(zˆ)∗y∗+N(xˆ,K)⊂D∗N(xˆ,ˆy)y∗ .

Proof It immediately follows from Theorem . and the proof of Theorem .. Similarly to Corollary . we have the following result.

Corollary . Let K be a compact set,xˆ∈K,yˆ∈N(xˆ),andzˆ∈M(xˆ,yˆ).

(i) IfFis Fréchet diﬀerentiable atzˆ,andMadmits a local upper Lipschitzian selection at(xˆ,ˆy,zˆ),then,for anyy∗∈Y∗satisfyingF(zˆ)∗y∗=∇F(zˆ)∗(y∗(xˆ–zˆ)∗)and

sup_s_∈_S_\{__}y∗_s,s=:v< ,we have

ˆ

D∗N(xˆ,yˆ)y∗ =F(zˆ)∗y∗+Nˆ(xˆ,K).

(ii) In addition to the conditions in case(i),ifKis regular atxˆ,thenNis M-regular at

(xˆ,ˆy),and for anyy∗∈Y∗satisfyingF(zˆ)∗y∗=∇F(zˆ)∗(y∗(xˆ–zˆ)∗)and

D∗_MN(xˆ,ˆy)y∗ =F(ˆz)∗y∗+N(xˆ,K).

(iii) In addition to the conditions in case(ii),ifKhas a compact base,thenNis N-regular at(xˆ,yˆ),and for anyy∗∈Y∗satisfyingF(zˆ)∗y∗=∇F(zˆ)∗(y∗(xˆ–zˆ)∗)and

D∗_NN(xˆ,yˆ)y∗ =F(zˆ)∗y∗+N(xˆ,K).

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(ii) SinceKis regular atxˆ, we haveNˆ(xˆ,K) =N(xˆ,K). Then

D∗_MN(xˆ,ˆy)y∗ ⊂F(ˆz)∗y∗+N(xˆ,K)

=F(ˆz)∗y∗+Nˆ(xˆ,K)

=Dˆ∗N(xˆ,yˆ)y∗

⊂D∗_MN(xˆ,yˆ)y∗ .

Thus, we have

D∗_MN(xˆ,ˆy)y∗ =Dˆ∗N(xˆ,yˆ)y∗ =F(zˆ)∗y∗+N(xˆ,K),

which impliesNisM-regular at (xˆ,yˆ).

(iii) It can be proved similar to case (ii). This completes the proof.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{College of Mathematics and Statistics, Nanyang Normal University, Nanyang, Henan 473061, China.}2_{College of Statistics} and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China.

Acknowledgements

The ﬁrst author was supported by the Scientiﬁc Research Fund for Advanced Talents of Nanyang Normal University. The second author was supported by the National Natural Science Foundation of Yunnan Province (No. 2014FD023). The authors would like to thank the anonymous referees for their valuable comments and suggestions, which helped to improve the paper.

Received: 13 January 2015 Accepted: 5 June 2015

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