Error bounds of regularized gap functions for weak vector variational inequality problems

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R E S E A R C H

Open Access

Error bounds of regularized gap functions for

weak vector variational inequality problems

Minghua Li

*

*_{Correspondence:}

[email protected] College of Science, Northwest A&F University, xinong road, Yangling, Shaanxi, China

Abstract

In this paper, by the nonlinear scalarization method, a global error bound of a weak vector variational inequality is established via a regularized gap function. The result extends some existing results in the literature.

MSC: 49K40; 90C31

Keywords: error bound; regularized gap function; weak vector variational inequality

1 Introduction

Throughout this paper, letK be a closed convex subset of an Euclidean spaceRn_and

F:Rn_→_B₍_Rn_,_Rm_{) be a continuously diﬀerentiable mapping. We consider a weak vector} variational inequality (WVVI) of ﬁndingx∗∈Ksuch that

Fx∗,x–x∗∈/–intC, ∀x∈K,

whereC⊆Rm_{is a closed convex and pointed cone with nonempty interior}_int_C_{. (WVVI)} was ﬁrstly introduced by Giannessi []. It has been shown to have many applications in vector optimization problems and traﬃc equilibrium problems (e.g., [, ]).

Error bounds are to depict the distance from a feasible solution to the solution set, and have played an important role not only in sensitivity analysis but also in convergence anal-ysis of iterative algorithms. Recently, kinds of error bounds have been presented for weak vector variational inequalities in [–]. By using a scalarization approach of Konnov [], Li and Mastroeni [] established the error bounds for two kinds of (WVVIs) with set-valued mappings. By a regularized gap function and a D-gap function for a weak vector variational inequality, Charitha and Dutta [] obtained the error bounds of (WVVI), re-spectively. Recently, in virtue of the regularized gap functions, Sun and Chai [] studied some error bounds for generalized (WVVIs). By using the image space analysis, Xu and Li [] got a gap function for (WVVI). Then, they established an error bound for (WVVI) without the convexity of the constraint set. These papers have a common characteristic: the solution set of (WVVI) is a singleton [, ]. Even though the solution set of (WVVI) is not a singleton [, ], the solution set of the corresponding variational inequality (VI) is a singleton, when their results reduce to (VI).

In this paper, by the nonlinear scalarization method, we study a global error bound of (WVVI). This paper is organized as follows. In Section , we establish a global error bound

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of (VI) via the generalized gap functions. In Section , we discuss a global error bound of (WVVI) by the nonlinear scalarization method.

2 A global error bound of (VI)

Let h : Rn _→ _R _{∪ {}₊_∞} _{be a proper lower semicontinuous function, and let} _S ₌

{x∈Rn_|_h₍_x₎_≤__}_._h_{has a global error bound if there exists}_τ_{>  such that}

d(x,S)≤τh(x)+, ∀x∈X,

whereh(x)+:=max{h(x), }andd(x,S) :=inf{x–s|s∈S}ifSis nonempty andd(x,S) = +∞ifSis empty.f :Rn_→_Rn_{is said to be coercive on}_K_if

lim

x∈K,x→+∞

f(x),x–y

x = +∞, ∀y∈K.

f:Rn_→_Rn_{is said to be strongly monotone on}_Rn_{with the modulus}_λ_{>  if}

f(x) –fx,x–x≥λx–x, ∀x,x∈Rn.

In this section, we establish a global error bound of (VI) of ﬁndingx∈Ksuch that

f(x),y–x≥, ∀y∈K,

wheref :Rn_→_Rn_{is a continuously diﬀerentiable mapping.}

To study the error bound of (VI), we need to construct a class of merit functions which were made to reformulate (VI) as an optimization problem; see [–]. One of such func-tions is a generalized regularized gap function [] deﬁned by

fγ(x) := –inf

y∈K

f(x),y–x+γ ϕ(x,y), ∀x∈Rn,γ > , ()

whereϕ:Rn_×_Rn_→_R_{is a real-valued function with the following properties:}

(P) ϕis continuously diﬀerentiable onRn_×_Rn_.

(P) ϕ(x,y)≥,∀x,y∈Rn_{and the equality holds if and only if}_x₌_y.

(P) ϕ(x,·)is uniformly strongly convex onRn_{with the modulus}_β_{> }_{in the sense that} ϕ(x,y) –ϕ(x,y)≥

∇ϕ(x,y),y–y

+βy–y, ∀x,y,y∈Rn,

where∇ϕdenotes the partial derivative ofϕwith respect to the second variable. (P) ∇ϕ(·,y)is uniformly Lipschitz continuous onRn_{with the modulus}_α_,_i.e._{, for all}

x∈Rn_,

_∇_ϕ(x,y) –∇ϕ(x,y)≤αy–y, ∀y,y∈Rn.

Now we recall some properties ofϕin ().

Proposition . The following statements hold for each x,y∈K:

(i) ∇ϕ(x,y),u ≤αx–yu,∀u∈span(K–x).

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(iii) ϕ(x,y) –∇ϕ(x,y),y–x ≥–(α–β)x–y.

(iv) ∇ϕ(x,y) = if and only ifx=y.

Proof Parts (i)-(iii) are taken from [, Lemma .] and part (iv) is from [, Lemma .].

Remark . In light of (ii) in Proposition ., it holds true thatα≥β.

Then we list some basic properties of the generalized regularized gap functionfγ.

Proposition . The following conclusions are valid for(VI).

(i) For everyx∈Rn_,_{there exists a unique vector}_yϕ

γ(x)∈Kat which the inﬁmum in()

is attained,i.e.,

fγ(x) = –

f(x),yϕγ(x) –x

–γ ϕx,yϕγ(x)

.

(ii) fγ is a gap function of(VI).

(iii) x=yϕ

γ(x)if and only ifxis a solution of(VI).

(iv) fγ is continuously diﬀerentiable onRnwith

∇fγ(x) = –∇f(x)

yϕγ(x) –x

+f(x) –γ∇ϕ

x,yϕγ(x)

.

(v) yϕ

γ andfγ are both locally Lipschitz onRn.

(vi) Iff is coercive onK,then(VI)has a nonempty compact solution set. (vii) fγ(x)≥βγyγϕ(x) –x,∀x∈K.

Proof Parts (i)-(iv) are from [], part (v) from [, Lemma .] and part (vi) from [, Proposition ..].

It follows from (ii) and (iii) that we only need to prove (vii) forx∈K\S. Sinceyϕ γ(x) is

the minimizer of the function

G(·) :=f(x),·–x+γ ϕ(x,·) onK, the ﬁrst-order optimality condition implies that

∇Gyϕγ(x)

,y–yϕγ(x)

≥, ∀y∈K.

Lettingy=x, we get

∇Gyϕγ(x)

, –yϕγ(x) +x

≥,

i.e.,

f(x) +γ∇ϕ

x,yϕγ(x)

,yϕγ(x) –x

≤.

It follows from (P) that the mappingGis strongly convex onRn_{with the modulus}_β_{> ,}

i.e.,∀x,y,y∈Rn

G(x,y) –G(x,y)≥f(x) +γ∇ϕ(x,y),y–y

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Lettingy=xandy=yϕγ(x), byfγ(x) = –G(x,yϕγ(x)), we obtain

fγ(x)≥

f(x) +γ∇ϕ

x,yϕγ(x)

,x–yϕγ(x)

+βγyϕγ(x) –x

 .

Thus, one hasfγ(x)≥βγyϕγ(x) –x.

Theorem . Let f be coercive on K andγ(α– β) <μ.Assume thatϕsatisﬁes (P) ∇ϕ(x,yϕγ(x)) +∇ϕ(x,yϕγ(x)),yϕγ(x) –x ≥,∀x∈K.

Suppose further that the following condition holds:

μ:=inf d,∇f(x)dx∈K\S,d= y

ϕ γ(x) –x

yϕγ(x) –x

> , ()

where S is the solution set of(VI).Thenfγ has a global error bound with the modulus

max 

√

βγ μ+ γ β–γ α,

√βγ βγ

.

Proof It follows from (vi) of Proposition . thatSis a nonempty compact set ofK. Ifx∈S, then the assertion obviously holds. Letx∈K\S. Thenfγ(x) > . For brevity, we denote

w:=yϕ

γ(x) –xandd:=ww. It follows from [, Theorem .] that we only need to prove

∇fγ(x)d≤–min

μ+ γ β–γ α

√βγ ,

√

βγ 

. ()

It follows from (iv) of Proposition . that

∇fγ(x)w=

–∇f(x)w+f(x) –γ∇ϕ

x,yϕγ(x)

,w

=–∇f(x)w,w+f(x),w+γ ϕx,yϕγ(x)

–γ∇ϕ

x,yϕ γ(x)

,w+ϕx,yϕ γ(x)

=–∇f(x)w,w–fγ(x) –γ

∇ϕ

x,yϕ γ(x)

,w+ϕx,yϕ γ(x)

.

By (P) and (), we have

∇fγ(x)w≤–μw–fγ(x) –γ

–∇ϕ

x,yϕγ(x)

,w+ϕx,yϕγ(x)

.

It follows from (iii) of Proposition . that

∇fγ(x)w≤–μw–fγ(x) +γ(α–β)w.

Thus,

∇fγ(x)d=

∇fγ(x)w_w

fγ(x)

≤–[μ–γ(α–β)]w

fγ(x)

–

fγ(x)

w . ()

In light of () and (vii) of Proposition ., we have

∇fγ(x)d≤–

[μ–γ(α–β)]w

fγ(x)

–

√

βγ

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Ifμ<γ(α–β), then it follows fromγ(α– β) <μthat

∇fγ(x)d≤

[γ(α–β) –μ]

√βγ –

√

βγ

 =

γ α–μ– γ β

√βγ < . ()

Ifμ≥γ(α–β), then

∇fγ(x)d≤–

√

βγ

 . ()

Hence, () follows from () and (). The proof is complete.

Now we use two examples to show that () cannot be dropped and that Theorem . is applicable, respectively.

Example . ConsiderK=R,ϕ(x,y) =_|x–y|,γ =_ andf(x) =x. Then we can easily get thatα= β= ,yϕ

γ(x) =x– f(x),fγ(x) =xandS={}. It is clear thatf is coercive

onKandμ= . Thus, () does not hold. Moreover, it is obvious thatfγ does not have a

global error bound.

Example . ConsiderK= [, +∞),ϕ(x,y) = _|x–y|_,_γ _{=  and}_f₍_x_{) =}_x_{. Then we can} easily get thatα= β= ,∇f(x) = ,yϕ

γ(x) = ,fγ(x) =_x andS={}. It is clear thatf is

coercive onKand () holds. Thus, it follows from Theorem . thatfγ has a global error

bound.

By [, Proposition .(ii)] Huang and Ng [, Theorem .] have obtained the following conclusion. Now we give a slightly diﬀerent proof by Theorem ..

Corollary . Let f be strongly monotone on Rnwith the modulusλ> andγ(α– β) <λ.

Assume thatϕsatisﬁes(P).Thenfγ has a global error bound with the modulus

max 

√

βγ λ+ γ β–γ α,

√βγ βγ

.

Proof Letx∈K\Sandw=yϕ

γ(x) –x. Sincef is continuously diﬀerentiable, then

f(x+tw) =f(x) +t∇f(x),w+o(t),

where o(t)_t → ast→. Sincef is strongly monotone with the modulusλ, one has

f(x+tw) –f(x),tw≥λtw,

which implies that

w,∇f(x)w≥λw.

Thus, () holds. Moreover, the strong monotonicity off implies the coerciveness off (cf.

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3 A global error bound of (WVVI)

In this section, by the nonlinear scalarization method and by Theorem ., we discuss a global error bound of (WVVI). The dual cone of C is deﬁned by C∗ :={ξ ∈Rm _:

ξ,z ≥,∀z∈C}. For eachξ∈Rm,ξ:=sup{|ξ,z|:z ≤}, whereξ,zdenotes the value ofξ atz. Lete∈intCandB∗_e :={ξ ∈C∗:ξ,e= }. It is well known thatB∗_e is a compact convex base ofC∗.

Lemma .[]

S⊃

ξ∈C∗\{} Sξ =

ξ∈B∗e Sξ,

where Sξ:={x∈K:

m

i=ξiFi(x∗),x–x∗ ≥,∀y∈K}and S is the solution set of(WVVI). Recall the generalized regularized gap function for (WVVI) which is deﬁned by

φγ(x) :=min ξ∈B∗e

fγ(x,ξ),

wherefγ(x,ξ) =maxy∈K{

m

i=ξiFi(x),x–y–γ ϕ(x,y)}. Whenφ(x,y) =_x–y, the gen-eralized regularized gap function reduces to the regularized gap function which was de-ﬁned in [].

Theorem . Letγ(α– β) <minξ∈Be∗μξ.Assume thatϕ satisﬁes(P).For eachξ ∈B ∗

e,

suppose thatξ◦F is coercive on K,and that the following condition holds:

μξ :=inf d, (ξ◦ ∇F)(x)dx∈K\S,d=

yϕ γ(x) –x

yϕγ(x) –x

> . ()

Thenφγ has a global error bound with the modulus

max max

ξ∈B∗e

√βγ μξ+ γ β–γ α

,

√

βγ βγ

.

Proof It follows from (vi) of Proposition . thatSξ is a nonempty compact set ofKfor

eachξ∈B∗_e. Ifx∈S, then the assertion obviously holds. Letx∈K\S. Thenφγ(x) >  and

there existsξ∈B∗e such thatfγ(x,ξ) =φγ(x). It follows from Theorem . that

d(x,Sξ)≤τξ

fγ(x,ξ),

whereτξ=max{ 

√

βγ μξ+γ β–γ α,

√βγ

βγ }. Thus, by Lemma ., one has

d(x,S)≤d(x,Sξ)≤τξ·

fγ(x,ξ)≤max ξ∈B∗e

τξ·

φγ(x).

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Remark . IfFiis strongly monotone with the modulusλifori= , , . . . ,mandC=Rm+, it follows from [, Proposition .] that

d, (ξ◦ ∇F)(x)d≥λd, ∀d∈Rn,ξ∈B∗_e.

Moreover, the strong monotonicity ofFiimplies the coerciveness ofFi(cf.[, Remark .]) and that (VI) has a unique solution (cf.[, Theorem ..]). Thus, by Theorem ., we get thatφγ has a global error bound. Hence, our results extend those of [, Theorem .].

Competing interests

The author declares that he has no competing interests.

Acknowledgements

This research was supported by the Natural Science Foundation of Shaanxi Province, China (Grant number: 2014JQ1023).

Received: 20 June 2014 Accepted: 18 August 2014 Published: 2 September 2014 References

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doi:10.1186/1029-242X-2014-331

Cite this article as:Li:Error bounds of regularized gap functions for weak vector variational inequality problems.