Scalar gap functions and error bounds for generalized mixed vector equilibrium problems with applications

R E S E A R C H

Open Access

Scalar gap functions and error bounds for

generalized mixed vector equilibrium

problems with applications

Wenyan Zhang

_{, Jiawei Chen}

_{, Shu Xu}

_{and Wen Dong}

*_{Correspondence:}

[email protected] 2_{School of Mathematics and}

Statistics, Southwest University, Chongqing, 400715, China Full list of author information is available at the end of the article

Abstract

It is well known that equilibrium problems are very important mathematical models and are closely related with fixed point problems, variational inequalities, and Nash equilibrium problems. Gap functions and error bounds which play a vital role in algorithms design, are two much-addressed topics of vector equilibrium problems. This paper is devoted to studying the scalar-valued gap functions and error bounds for the generalized mixed vector equilibrium problem (GMVE). First, a scalar gap function for (GMVE) is proposed without any scalarization methods, and then error bounds of (GMVE) are established in terms of the gap function. As applications, error bounds for generalized vector variational inequalities and vector variational

inequalities are derived, respectively. The main results obtained are new and improve corresponding results of Charitha and Dutta (Pac. J. Optim. 6:497-510, 2010) and Sun and Chai (Optim. Lett. 8:1663-1673, 2014).

Keywords: error bound; scalar gap function; generalized mixed vector equilibrium

1 Introduction

Equilibrium problems, which were firstly studied by Blum and Oettli [], provide an uni-fied framework for fixed point problems, variational inequality, complementarity prob-lems, and optimization problems. It is well known that the vector equilibrium problem is a vital extension of equilibrium problems, which contain vector variational inequality, vector complementarity problems, and vector optimization problems as special cases. In the past decades, various kinds of vector equilibrium problems and their applications have been introduced and studied; see [–] and the references therein. Recently, Changet al. [] and Kumamet al.[] researched a generalized mixed equilibrium problem,

F(x,y) +A(x),y–x+g(y) –g(x)≥, ∀y∈K.

It is very general in the sense that it includes fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases.

Error bounds, which play a critical role in algorithm design, can be used to measure how much the approximate solution fails to be in the solution set and to analyze the con-vergence rates of various methods. Recently, kinds of error bounds have been presented

for variational inequalities in [–]. Results for error bounds have been established for a weak vector variational inequality (WVVI) in [–, ]. Xu and Li [] obtained error bounds for a weak vector variational inequality with cone constraints by using a method of image space analysis. By using a scalarization approach of Konnov [], Li and Mastroeni [] established error bounds for two kinds of (WVVI) with set-valued mappings. By a regularized gap function and a D-gap function, Charitha and Dutta [] used a projec-tion operator method to obtain error bounds of (WVVI), respectively. Sun and Chai [] studied some error bounds for generalized vector variational inequalities in virtue of the regularized gap functions. Very recently, a global error bound of a weak vector variational inequality was established by the nonlinear scalarization method in Li [].

However, to the best of our knowledge, an error bound of the generalized mixed vector equilibrium problem (GMVE) has never been investigated. In this paper, motivated by ideas in Sun and Chai [] and Yamashitaet al.[], we introduce a scalar gap function for (GMVE). Then an error bound of (GMVE) is presented. As an application of an error bound for (GMVE), we also get error bounds of (GVVI) and (VVI), respectively.

This paper is organized as follows: In Section , we first recall some basic definitions. In Section , we introduce scalar gap functions for (GMVE), (GVVI), and (VVI). By using these gap functions, we obtain some error bound results for (GMVE), (GVVI), and (VVI), respectively.

2 Mathematical preliminaries

Throughout this paper, letRn_{be then-dimensional Euclidean space and denoteR}n +={x= (x,x, . . . ,xn) :xj≥,j= , , . . . ,n}, the norms of all finite dimensional spaces be denoted by · , the inner products of all finite dimensional spaces be denoted by·,·. LetK⊆Rn be a nonempty closed convex set. Let Ai:Rn_→Rn _{(i= , , . . . ,m) be a vector-valued} mapping,Fi:Rn×Rn→Randgi:Rn→R(i= , , . . . ,m) be real-valued functions. For abbreviation, we put

A:= (A,A, . . . ,Am), F:= (F,F, . . . ,Fm), g:= (g,g, . . . ,gm)

and for anyx,v∈Rn

A(x),v:=A(x),v

,A(x),v

, . . . ,Am(x),v

.

In this paper, we consider the generalized mixed vector equilibrium problem (GMVE) of findingx∈Ksuch that

F(x,y) +A(x),y–x+g(y) –g(x) /∈–intRm₊, ∀y∈K. ()

Denote bySGMVEthe solution set of (GMVE). Ifm= , our problem is to findx∈Ksuch that

F(x,y) +A(x),y–x+g(y) –g(x)≥, ∀y∈K. ()

IfF= , our problem is to findx∈Ksuch that

A(x),y–x+g(y) –g(x) /∈–intRm₊, ∀y∈K. () Then this problem reduces to a generalized vector variational inequality problem (GVVI) [].

In the case ofF=  andg= , (GMVE) is a vector variational inequality problem (VVI), introduced and studied by Giannessi [], findingx∈Ksuch that

A(x),y–x∈/–intRm₊, ∀y∈K. ()

In the case ofA≡ andg= , then (GMVE) is a vector equilibrium problem, finding x∈Ksuch that

F(x,y) /∈–intRm₊, ∀y∈K. ()

In the case ofA≡ andF= , (GMVE) is a vector optimization problem, findingx∈K such that

g(y) –g(x) /∈–intRm₊, ∀y∈K. ()

For i = , , . . . ,m, we denote the generalized mixed vector equilibrium problems (GMVE) associated withFi,Aiandgias (GMVE)i, the generalized vector variational in-equality problems (GVVI) associated withAiandgias (GVVI)i, and the vector variational inequality problems (VVI) associated withAias (VVI)i, respectively. The solution sets of (GMVE)i_{, (GVVI)}i_{, and (VVI)}i_{will be denoted byS}i

GMVE,SGVVIi , andSVVIi , respectively. In the paper, we intend to investigate gap functions and error bounds of (GMVE), (GVVI), and (VVI). We shall recall some notations and definitions, which will be used in the sequel.

Definition . A real-valued functionf :Rn_{→Ris convex (resp. concave) overR}n_{, if}

fλx+ ( –λ)y≤λf(x) + ( –λ)f(y)

resp.fλx+ ( –λ)y≥λf(x) + ( –λ)f(y), for everyx,y∈Rn_{andλ∈[, ].}

Definition . A vector-valued functionh:Rn_→Rn_{is strongly monotone overR}n_with modulusκ> , if for anyx,y∈Rn,

h(y) –h(x),y–x≥κy–x.

Definition . A real-valued functionϑ:Rn→Ris said to be a scalar-valued gap func-tion of (GMVE) (resp. (GVVI) and (VVI)), if it satisfies the following condifunc-tions:

(i) ϑ(x)≥, for anyx∈K,

3 Main results

Notice that Sun and Chai [] introduced a new scalar-valued gap function of (GVVI) without any scalarization approach; the gap function discussed in [] is simpler from the computational view. In terms of an approach due to Sun and Chai [] and Yamashitaet al.[], we construct the functionϑα:K→Rforα> ,

ϑα(x) :=sup

y∈K

min

≤i≤m

–Fi(x,y) +Ai(x),x–y+gi(x) –gi(y)–αϕ(x,y) , ()

ψα(x) :=sup

y∈K

min

≤i≤m

Ai(x),x–y+gi(x) –gi(y)–αϕ(x,y) , ()

and

φα(x) :=sup

y∈K

min

≤i≤m

Ai(x),x–y–αϕ(x,y)}, ()

respectively, whereFi,ϕ:Rn_×Rn_{→R, i= , , . . . ,mare real-valued functions. In the} following, letϕbe a continuously differentiable function, which has the following property with the associated constantsγ,β> .

(P) For allx,y∈Rn_,

βx–y≤ϕ(x,y)≤(γ –β)x–y, γ ≥β> . ()

For example, letκ>  and letϕ:Rn_×Rn_{→Rbe defined by}

ϕ(x,y) =κx–y, x,y∈Rn.

Thenϕsatisfies condition (P), withγ = κandβ=κ.

For anyi= , , . . . ,m, we suppose thatFisatisfies the following conditions: (A) Fiis a convex function about the second variable onRn_×Rn_. (A) Fi(x,y) = ,∀x,y∈Rn_{, if and only ifx=y.}

(A) For anyx,y,z∈K,Fi(x,y) +Fi(y,z)≤Fi(x,z).

Theorem . If Fi:Rn_×Rn_{→Ris convex about the second variable and giis convex over} Rn_{for any i= , , . . . ,m,then the functionϑ}

α,withα> ,defined by()is a gap function

for(GMVE).

Proof (i) It is clear that for anyx∈K,

ϑα(x) :=sup

y∈K

min

≤i≤m

–Fi(x,y) +

Ai(x),x–y

+gi(x) –gi(y)

–αϕ(x,y) ≥

follows simply by settingx=yin the right hand side of the expression forϑα(x).

(ii) If there existsx∈Ksuch thatϑα(x) = , then

min

≤i≤m

–Fi(x,y) +

Ai(x),x–y

+gi(x) –gi(y)

For arbitraryx∈K andκ∈(, ), lety=x+κ(x–x). SinceKis convex, we gety∈K and

min

≤i≤m

–Fi

x,x+κ(x–x)

+Ai(x),x–

x+κ(x–x)

+gi(x) –gi

x+κ(x–x)

≤αϕx,x+κ(x–x)

.

SinceFiis convex overRn_×Rn_{about the second variable andgiis convex overR}n_{for any} i= , , . . . ,m, we have

min

≤i≤m

–κFi(x,x) – ( –κ)Fi(x,x) +

Ai(x), ( –κ)(x–x)

+gi(x) –κgi(x) – ( –κ)gi(x)

≤ min

≤i≤m

–Fi

x,x+κ(x–x)

+Ai(x), ( –κ)(x–x)

+gi(x) –gi

x+κ(x–x)

≤αϕx,x+κ(x–x)

. ()

For the functionϕ(x,y), by using (), we have

ϕx,x+κ(x–x)

≤( –κ)(γ–β)x–x. ()

By the property (A) of the functionFi, we obtainFi(x,x) = . Hence, from () and (), we get

( –κ) min

≤i≤m

–Fi(x,x) +

Ai(x),x–x

+gi(x) –gi(x)

≤α(γ–β)( –κ)x–x.

So,

min

≤i≤m

–Fi(x,x) +

Ai(x),x–x

+gi(x) –gi(x)

≤α(γ–β)( –κ)x–x.

Taking the limit asκ→, we obtain

min

≤i≤m

–Fi(x,x) +

Ai(x),x–x

+gi(x) –gi(x)

≤.

Then, for anyx∈K, there exists ≤i≤msuch that

–Fi(x,x) +

Ai(x),x–x

+gi(x) –gi(x)≤.

This means that

F(x,x) +

Ai(x),x–x

+g(x) –g(x) /∈–intRm+, ∀x∈K.

Conversely, ifx∈SGMVE, then there exists ≤i≤msuch that

Fi(x,y) +

Ai(x),y–x

+gi(y) –gi(x)≥ for anyy∈K.

This means that

min

≤i≤m

–Fi(x,y) +

Ai(x),x–y

+gi(x) –gi(y)

≤ for anyy∈K.

So,

ϑα(x)≤.

Sinceϑα(x)≥ for anyx∈K,

ϑα(x) = .

This completes the proof.

By a similar method, we conclude the following results for (GVVI) and (VVI), respec-tively.

Corollary . The functionψα,withα> ,defined by()is a gap function for(GVVI).

Corollary . The functionφα,withα> ,defined by()is a gap function for(VVI).

Now, by using the gap functionϑα(x), we obtain an error bound result for (GMVE).

Theorem . Assume that each Aiare strongly monotone over K with the modulusκi> . Assume Fiis convex about the second variable overRn×Rnand giis convex overRnfor any i= , , . . . ,m.Further assume thatm_i=S_GMVEi =∅.Moreover,letκ=min≤i≤mκiand

α> be chosen such thatκ>α(γ–β),whereγ ≥β> are constants associated with the functionϕ.Then for any x∈K,

d(x,SGMVE)≤ 

κ–α(γ –β)

ϑα(x),

where d(x,SGMVE)denotes the distance from the point x to the solution set SGMVE.

Proof By (), we get

ϑα(x)≥ min

≤i≤m

–Fi(x,y) +Ai(x),x–y+gi(x) –gi(y)–αϕ(x,y) for anyy∈K.

Sincem_i=Si

GMVE=∅, all (GMVE)ihave the same solution. Without loss of generality, we can assume that the same solution isx∈K. Obviously,x∈SGMVEand

ϑα(x)≥ min

≤i≤m

–Fi(x,x) +

Ai(x),x–x

+gi(x) –gi(x)

Without loss of generality, we assume that

–F(x,x) +

A(x),x–x

+g(x) –g(x)

= min

≤i≤m

–Fi(x,x) +

Ai(x),x–x

+gi(x) –gi(x)

.

SinceAis strongly monotone, we obtain

ϑα(x)≥–F(x,x) +

A(x),x–x

+g(x) –g(x) +κx–x–αϕ(x,x).

It follows from the property (P) of the functionϕthat

ϑα(x)≥–F(x,x) +

A(x),x–x

+g(x) –g(x)

+κx–x–α(γ –β)x–x. ()

Byx∈SGMVE, we get

F(x,x) +

A(x),x–x

+g(x) –g(x)≥. ()

For the functionF, by using (A), we get from () and ()

ϑα(x) =ϑα(x) +F(x,x)≥ϑα(x) +F(x,x) +F(x,x)≥

κ–α(γ –β)x–x. Namely,

ϑα(x)≥

κ–α(γ –β)x–x. Then

x–x ≤



κ–α(γ–β)

ϑα(x),

which means that

d(x,SGMVE)≤ 

κ–α(γ –β)

ϑα(x).

This complete the proof.

The following example shows that, in general, the conditions of Theorem . can be achieved.

Example . Letn= ,m= ,K⊆R, andK= [–, ]. DefineA,A,g,g:R→R, F : R×R→R,F:R×R→Rby

A(x) =x, A(x) =x+ x, g(x) = x, g(x) = x,

Then

F(x,y) =–x+y, –x+ y, A(x) =x,x+ x, g(x) =x, x. Obviously,F(x,y) andF(x,y) are convex about the second variable, respectively.Aand Aare strongly monotone overKwith the modulusκ=  andκ= , respectively. More-over,gandgare convex overR. On the other hand, by direct calculations, we have



i=

Si_GMVE={}.

Thus, the conditions of Theorem . are satisfied.

Similarly, by using gap functionsψαandφα, we can also obtain error bound results for

(GVVI) and (VVI), respectively.

Corollary . Assume that each Aiare strongly monotone over K with the modulusκi> . If gi is convex overRn_{for any i= , , . . . ,m.Further assume that}m

i=SiGVVI=∅. More-over,letκ=min≤i≤mκiandα> be chosen such thatκ>α(γ –β),whereγ ≥β> are constants associated with the functionϕ.Then,for any x∈K,

d(x,SGVVI)≤ 

κ–α(γ–β)

ψα(x),

where d(x,SGVVI)denotes the distance from the point x to the solution set SGVVI.

Corollary . Assume that each Aiare strongly monotone over K with the modulusκi> . Further assume thatm_i=Si_VVI=∅.Moreover,letκ=min≤i≤mκiandα> be chosen such thatκ>α(γ–β),whereγ ≥β> are constants associated with the functionϕ.Then,for any x∈K,

d(x,SVVI)≤



κ–α(γ–β)

φα(x),

where d(x,SVVI)denotes the distance from the point x to the solution set SVVI.

Remark . (i) In [], there exist some mistakes in the proof of Theorem ., which lead to the requirement of Lipschitz properties ofgi,i= , , . . . ,n. Hence, we give the modified error bound for (GVVI) in Corollary . without Lipschitz assumption.

(ii) In [], Charitha and Dutta established error bounds for (VVI) by using the pro-jection operator method and strongly monotone assumptions, whereas it seems that our method is more simple from the computational view since there are not any scalarization parameters.

(iii) Under the conditions of Theorem ., the strong assumption thatm_i=Si_GMVE=∅ shows thatSGMVEis a singleton set but not a general set. In fact, bym_i=Si_GMVE=∅, there existsx∈Ksuch thatx∈m_i=Si

GMVE, namely, for everyi= , , . . . ,m,

It is clear thatx∈SGMVE. IfSGMVEis not a singleton set, there existsx∈SGMVEwithx=x. Therefore, there existsj∈ {, , . . . ,m}such that

Fj

x,y+Aj

x,y–x+gj(y) –gj

x≥, ∀y∈K. ()

Thus, from (), we have

Fjx,x+Aj(x),x–x+gjx–gj(x)≥. () From (), we have

Fjx,x+Ajx,x–x+gj(x) –gjx≥. () According to () and (), we get

Fjx,x+Fjx,x+Aj(x),x–x+Ajx,x–x≥. () However, by the properties (A) and (A) of the functionFj,

Fjx,x+Fjx,x≤. ()

AsAjis strongly monotone, we have

Aj(x) –Ajx,x–x≥κx–x> . () By combining () and (), we have

Fjx,x+Fjx,x+Aj(x) –Ajx,x–x< . () This, however, contradicts ().

Now we ask: How do we establish error bounds forSGMVE in terms of the gap func-tionϑα, under mild assumptions, such thatSGMVEneed not be a singleton set in general? This problem may be interesting and valuable in vector optimization.

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_{The School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China.} 2_{School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China.}3_{Education Committee of Banan}

District, Chongqing, 401320, China.

Acknowledgements

The authors would like to thank the associated editor and the two anonymous referees for their valuable comments and suggestions, which have helped to improve the paper. This work was partially supported by the National Natural Science Foundation of China (Grant number: 11401487), the Fundamental Research Funds for the Central Universities (Grant numbers: SWU113037, XDJK2014C073), and the Scientific Research Fund of Sichuan Provincial Science and Technology Department (Grant number: 2015JY0237).

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