AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 596067 10.1155/2014/596067 596067 Research Article An Improvement of Global Error Bound for the Generalized Nonlinear Complementarity Problem over a Polyhedral Cone http://orcid.org/0000-0003-0510-2199 Sun Hongchun 1 Wang Yiju 2 Zhou Houchun 1 Li Shengjie 3 Xia Fu-quan 1 School of Sciences Linyi University Linyi Shandong 276005 China lyu.edu.cn 2 School of Management Science Qufu Normal University Rizhao Shandong 276800 China qfnu.edu.cn 3 College of Mathematics and Statistics Chongqing University Chongqing 401331 China cqu.edu.cn 2014 2942014 2014 27 02 2014 16 04 2014 29 4 2014 2014 Copyright © 2014 Hongchun Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider the global error bound for the generalized nonlinear complementarity problem over a polyhedral cone (GNCP). By a new technique, we establish an easier computed global error bound for the GNCP under weaker conditions, which improves the result obtained by Sun and Wang (2013) for GNCP.

1. Introduction

Let 𝒦={vRmAv0,Bv=0} be a polyhedral cone in Rm for matrices ARs×m, BRt×m, and let 𝒦 be its dual cone; that is, (1)𝒦={uRmu=Aλ1+Bλ2,λ1R+s,λ2Rt}. For continuous mappings, F,G:RnRm, the generalized nonlinear complementarity problem, abbreviated as GNCP, is to find vector x*Rn such that (2)F(x*)𝒦,G(x*)𝒦,F(x*)G(x*)=0. Throughout this paper, the solution set of the GNCP, denoted by X*, is assumed to be nonempty.

The GNCP is a direct generalization of the classical nonlinear complementarity problem and a special case of the general variational inequalities problem . The GNCP was deeply discussed  after the work in . It is well known that the global error bound is an important tool for theoretical analysis and numerical treatment for a mathematical problem [9, 10]. The global error bound estimation for GNCP with the mapping being γ-strongly G-monotone was discussed in , a global error bound estimation for GNCP with the mapping being γ-strongly monotone and Hölder continuous was established in , and a global error bound for the GNCP for the linear and monotonic case was also established in [6, 7].

This paper is a follow-up to [4, 5, 11], as in these papers we will establish the global error bound estimation of the GNCP under weaker conditions than that needed in [4, 5, 11]. Based on a new technique, we establish a global error bound for the GNCP in terms of an easier computed residual function. The results obtained in this paper can be taken as an improvement of the existing results for GNCP and variational inequalities problem [4, 5, 1113].

To end this section, we give some notations used in this paper. Vectors considered in this paper are taken in Euclidean space Rn equipped with the usual inner product, and the Euclidean 2-norm and 1-norm of vector in Rn are, respectively, denoted by · and ·1. We use R+n to denote the nonnegative orthant in Rn and use x+ and x- to denote the vectors composed by elements (x+)i:=max{xi,0}, (x-)i:=max{-xi,0} and 1in, respectively. For simplicity, we use (x;y) to denote vector (x,y), use I to denote the identity matrix with appropriate dimension, use x0 to denote a nonnegative vector xRn, and use dist(x,X*) to denote the distance from point x to the solution set X*.

2. Global Error Bound for the GNCP

First, we give the following definition used in the subsequent.

Definition 1.

The mapping F:RnRm is said to be γ-uniform p-function with respect to G:RnRm if there are constants c1>0 and γ>0 such that (3)max1im{[F(x)-F(y)]i[G(x)-G(y)]i}mmmmmc1G(x)-G(y)1+γ,mmmmmmmmmmmlx,yRn.

Remark 2.

Based on this definition, γ-uniform p-function with respect to G is weaker than γ-strongly G-monotonicity by Definition 1 in , and if (4)F(x)=Mx+p,G(x)=Nx+q with M,NRm×n, p,qRm, then the above definition is equivalent in which the matrix MN is a p-matrix . For example, let (5)M=(1-411),N=(1001),p=q=(00). By Theorem 2.1.15 in , it is easy to verify that MN is a p-matrix. However, letting x=(1;2), we note that xMNx=-1<0 which shows that MN is not positive definite; that is, F is not strongly monotonicity with respect to mapping G.

Now, we give some assumptions for our analysis based on Definition 1.

Assumption 3.

For mappings F,G and matrix A involved in the GNCP, we assume that

mapping F is γ-uniform p-function with respect to mapping G;

matrix A has full-column rank.

In the following, we give the conclusion established in .

Theorem 4.

A point x*Rn is a solution of the GNCP if and only if there exist λ1*Rs and λ2*Rt, such that (6)AF(x*)0,BF(x*)=0,λ1*0,(F(x*))G(x*)=0,G(x*)=Aλ1*+Bλ2*.

From Theorem 4, under Assumption 3 (A2), similar to discussion in , we can transform the system (6) into the following system in which neither λ1 nor λ2 is involved: (7)AF(x)0,BF(x)=0,(F(x))G(x)=0,UG(x)0,VG(x)=0, where (8)U={-AL-1B[(AAL-1-I)B]+[AAL-1-I]+AL-1},V={A{-AL-1B[(AAL-1-I)B]+[AAL-1-I]+AL-1}mmm+B[(AAL-1-I)B]+[AAL-1-I]-IA{-AL-1B[(AAL-1-I)B]+[AAL-1-I]+AL-1}}.

For the ease of description, let μ=F(x) and ν=G(x). Then, system (7) can be written as (9)Aμ0,Bμ=0,μν=0,Uν0,Vν=0, where the solution set of (9) is denoted by Ω*.

In the following, we give the error bound for a single quadratic function to reach our aims.

Lemma 5.

Let S1:={ωR2mf(ω)=0}. Then, one has (10)dist(ω,S1)τf(ω)1/2, where τ>0 is a constant, ω=(μ;ν), and f(ω)=μν.

Proof.

For any ωR2m, let μ=(μ1,μ2,,μm), ν=(ν1,ν2,,νm), and (11)ωi={μi1im,νim+1i2m. Set μi=ξi+ξm+i, νi=ξi-ξm+i, i=1,2,,m, and ξ=(ξ1,ξ2,,ξ2m). Obviously, this linear transformation is an invertible; that is, there exists an invertible matrix PR2m×2m such that ω=Pξ, and one has (12)f(ω)=μν=i=1mμiνi=i=1mξi2-i=m+12mξi2=:g(ξ). Without loss of generality, we assume f(ω)>0. Define (13)θ=(i=m+12mξi2g(ξ)+i=m+12mξi2)1/2=(i=m+12mξi2i=1mξi2)1/2,ξ¯i={θξi1im,ξim+1i2m.

It is easy to verify that 0θ1 and (14)g(ξ¯)=θ2i=1mξi2-i=m+12mξi2=(i=m+12mξi2i=1mξi2)i=1mξi2-i=m+12mξi2=i=m+12mξi2-i=m+12mξi2=0. Let ω¯=Pξ¯, and one has f(ω¯)=g(ξ¯)=0. Therefore, ω¯S1. Moreover, one has (15)ω-ω¯=Pξ-Pξ¯Pξ-ξ¯=P(i=12m(ξi-ξ¯i)2)1/2=P(i=1m(ξi-θξi)2)1/2=P(1-θ)(i=1mξi2)1/2=P(1-θ2)1+θ(i=1mξi2)1/2=P1+θ(1-i=m+12mξi2g(ξ)+i=m+12mξi2)(i=1mξi2)1/2=Pg(ξ)(1+θ)(i=1mξi2)(i=1mξi2)1/2=Pg(ξ)(1+θ)(i=1mξi2)1/2=Pg(ξ)(i=1mξi2)1/2+θ(i=1mξi2)1/2=Pg(ξ)(i=1mξi2)1/2+((i=m+12mξi2)/(i=1mξi2))1/2(i=1mξi2)1/2=Pg(ξ)(i=1mξi2)1/2+(i=m+12mξi2)1/2Pg(ξ)(i=1mξi2+i=m+12mξi2)1/2Pg(ξ)g(ξ)1/2=Pg(ξ)1/2=Pf(ω)1/2, where the third equality follows from the definition of ξ¯, the sixth and tenth equations are due to the definition of θ, respectively, the second inequality follows from the fact that (16)a1/2+b1/2(a+b)1/2,a,bR+, and the third inequality follows from the fact that (17)g(ξ)i=1mξi2+i=m+12mξi2. And, letting τ=P, then the desired result follows.

To establish a global error bound for GNCP, we also give the following result from  on the error bound for a polyhedral cone.

Lemma 6.

For polyhedral cone P={xRnD1x=d1,B1xb1} with D1Rl×n, B1Rm×n, d1Rl, and b1Rm, there exists a constant c2>0 such that (18)dist(x,P)c2[D1x-d1+(B1x-b1)+],xRn.

Before proceeding, we present the following definition introduced in  with constant δ=1.

Definition 7.

The mapping G:RnRm is said to be δ-strongly nonexpanding with a constant α>0, if G(x)-G(y)αx-yδ, where δ>0.

Now, we are at the position to state our main results in this paper.

Theorem 8.

Suppose that F is γ-uniform p-function with respect to mapping G with positive constants c1 and γ, respectively, and G is δ-strongly nonexpanding with positive constants α and δ, respectively. Then there exists constant ρ1>0 such that (19)dist(x,X*)ρ1{[F(x)G(x)]1/2[AF(x)]-+BF(x)mm+[UG(x)]-+VG(x)mm+[F(x)G(x)]1/2}2/(1+γ)δ,xRn.

Proof.

Using Lemma 5, for any ωR2m, there exists ω¯S1 such that (20)ω-ω¯τ[f(ω)]1/2, where S1 is defined in Lemma 5. Let (21)Ω={ωR2mA(I,0)ω0,B(I,0)ω=0,U(0,I)ω0,V(0,I)ω=0R2mA(I,0)ω}. From (9), we have Ω*=ΩS1. For convenience, we also let (22)Ψ(ω)=(-A(I,0)ω,-B(I,0)ω,-U(0,I)ω,-V(0,I)ω,B(I,0)ω,V(0,I)ω)+. Using Lemma 6, for any ωS1, there exists ω*Ω* such that (23)ω-ω*c3[(-A(I,0)ω)++(-U(0,I)ω)+mmmll+B(I,0)ω+V(0,I)ω]c3[(-A(I,0)ω)++(-U(0,I)ω)+mmmll+(B(I,0)ω)+mmmll+(-B(I,0)ω)++(V(0,I)ω)+mmmll+(-V(0,I)ω)+]c3{(-A(I,0)ω)+1+(-U(0,I)ω)+1mmmll+(B(I,0)ω)+1mmmll+(-B(I,0)ω)+1+(V(0,I)ω)+1mmmll+(-V(0,I)ω)+1}=c3Ψ(ω)1c32s+2t+2mΨ(ω), where c3 is a positive constant and the third and fourth inequalities follow from the fact that xx1nx, for all xRn.

Furthermore, (24)Ψ(ω)-Ψ(ω¯)=-V(0,I)ω¯,B(I,0)ω¯,V(0,I)ω¯)+(-A(I,0)ω,-B(I,0)ω,-U(0,I)ω,mmmmm-V(0,I)ω,B(I,0)ω,V(0,I)ω)+mmmm-(-A(I,0)ω¯,-B(I,0)ω¯,-U(0,I)ω¯,mmmmmm-V(0,I)ω¯,B(I,0)ω¯,V(0,I)ω¯)+=PR+2s+2t+2m{(-A(I,0)ω,-B(I,0)ω,-U(0,I)ω,mmmmmmmm-V(0,I)ω,B(I,0)ω,V(0,I)ω)}mmmm-PR+2s+2t+2m{(-A(I,0)ω¯,-B(I,0)ω¯,-U(0,I)ω¯,mmmmmmmmmm-V(0,I)ω¯,B(I,0)ω¯,V(0,I)ω¯)}PR+2s+2t+2m{(-A(I,0)ω,-B(I,0)ω,-U(0,I)ω,{(-A(I,0)ω,-B(I,0)ω,-U(0,I)ω,mmmll-V(0,I)ω,B(I,0)ω,V(0,I)ω)}mmml-{(-A(I,0)ω¯,-B(I,0)ω¯,-U(0,I)ω¯,mmmmm-V(0,I)ω¯,B(I,0)ω¯,V(0,I)ω¯)}A(I,0)ω-A(I,0)ω¯+2B(I,0)ω-B(I,0)ω¯mmm+U(0,I)ω-U(0,I)ω¯+2V(0,I)ω-V(0,I)ω¯(A(I,0)+2B(I,0)+U(0,I)mmm+2V(0,I))ω-ω¯, where the second equality follows from the fact that (25)min{a,b}=a-PR+(a-b),a,bR, and the first inequality is by nonexpanding property of projection operator. Combining (24), one has (26)Ψ(ω¯)Ψ(ω)+(A(I,0)+2B(I,0)mml+U(0,I)+2V(0,I))ω-ω¯. Combining (23) with (26), for any ωR2m, we have (27)ω-ω*ω-ω¯+ω¯-ω*ω-ω¯+σΨ(ω¯)ω-ω¯+σ(Ψ(ω)+(A(I,0)+2B(I,0)mmmmmmmmmm+U(0,I)mmmmmmmlllllll+2V(0,I))ω-ω¯)σΨ(ω)+[σ(A(I,0)+2B(I,0)mmmm+U(0,I)+2V(0,I))+1]ω-ω¯σΨ(ω)+[σ(A(I,0)+2B(I,0)+U(0,I)mmmm+2V(0,I))+1]τ[f(ω)]1/2η(Ψ(ω)+[f(ω)]1/2)η(Ψ(ω)1+[f(ω)]1/2)η(+V(0,I)ω1+[f(ω)]1/2(-A(I,0)ω)+1+(-U(0,I)ω)+1mmmm+B(I,0)ω1mmmm+V(0,I)ω1+[f(ω)]1/2)η(+mV(0,I)ω+[f(ω)]1/2s(-A(I,0)ω)++s(-U(0,I)ω)+mmmm+tB(I,0)ωmmmm+mV(0,I)ω+[f(ω)]1/2)c4([f(ω)]1/2(-A(I,0)ω)++(-U(0,I)ω)+mmmm+B(I,0)ωmmmm+V(0,I)ω+[f(ω)]1/2), where the second inequality follows from (23) with constants σ=c32s+2t+2m and ω=ω¯, the third inequality uses (26), the fifth inequality follows from (20), the sixth inequality follows from the fact that (28)η=max{σ,[σ(A(I,0)+2B(I,0)mmmmml+U(0,I)+2V(0,I))+1]τ}, the seventh and ninth inequalities follow from the fact that (29)xx1nx,xRn, and the last inequality follows by letting c4=ηmax{s,t,m,1}.

For any xRn, let ω=(μ,ν)=(F(x),G(x))R2m. Then there exists ω*=(μ*,ν*)=(F(x*),G(x*))Ω* such that (30)dist(1+γ)δ(x,X*)x-x*(1+γ)δ1α1+γG(x)-G(x*)1+γ1c1α1+γmax1in{[F(x)-F(y)]i[G(x)-G(y)]i}1c1α1+γF(x)-F(x*)G(x)-G(x*)12c1α1+γ{F(x)-F(x*)2+G(x)-G(x*)2}=12c1α1+γω-ω*212c1α1+γc42(B(I,0)ω+V(0,I)ω+[f(ω)]1/2(-A(I,0)ω)++(-U(0,I)ω)+mmmmmmmmmm+B(I,0)ωmmmmmmmmmm+V(0,I)ω+[f(ω)]1/2)212c1α1+γc42(+V(0,I)ω+[f(ω)]1/2(A(I,0)ω)-+(U(0,I)ω)-  mmmmmmmmmm+B(I,0)ωmmmmmmmmmm+V(0,I)ω+[f(ω)]1/2)212c1α1+γc42{μν1/2[Aμ]-+Bμ+[Uν]-+Vνmmmmmmmmmm+μν1/2}2, where the second inequality follows from Definition 7 with constant α>0, the third inequality follows from Definition 1 with constants c1>0 and γ>0, the fifth inequality follows from the fact that a2+b22ab, for all a,bR, and the sixth inequality is by (27). By (30) and letting ρ1={(1/(2c1α1+γ))c42}1/(1+γ)δ, then the desired result follows.

Remark 9.

Firstly, from remark of Definition 1, the conditions that F is γ-uniform p-function with respect to mapping G and G is δ-strongly nonexpanding in Theorem 8 are weaker than the conditions that F is γ-strongly G-monotone and G is strongly nonexpanding (i.e., δ=1) in Theorem 13 in . In addition, the result in Theorem 8 is stronger than that in Theorem 13 in . Thus, Theorem 8 is stronger than Theorem 13 in .

In the following, we also present an example to compare the condition in Theorem 8 in this paper and that in Theorem 13 in .

Example 10.

When 𝒦=R+2, F(x)=Mx, and G(x)=x, the (2) reduces to the linear complementarity problem (LCP) of finding vector x*Rn such that (31)F(x*)0,G(x*)0,F(x*)G(x*)=0, where M=(1-301).

It is easy to see that the solution set of the LCP X*={0}. In fact, (32)X*={xR2x0,Mx0,xMx=0}={3+52(x1,x2)R2x10,x20,x13x2,mmlx1=3-52x2orx1=3+52x2}={0}. Clearly, M is a P matrix FV. Thus, there exists constant τ>0 such that (33)max1i2{(x-y)i(Mx-My)i}τx-y2. For any x(ϵ):=(-ϵ;ϵ), ϵ0. By Theorem 8, with γ=1, δ=1, A=I, U=I, B=0, and V=0 and letting φ1(x):=[F(x)]-+[G(x)]-+[F(x)G(x)]1/2, we can obtain (34)x(ϵ)-0φ1(x(ϵ))=2ϵ4ϵ+ϵ+5ϵ25+5 as ϵ0. Thus, Theorem 8 provides a global error bound for this LCP.

However, letting x=(1;1), we note that xMx=-1<0 which shows that M is not positive definite, so the condition that F is strongly monotone in Theorem 13 in  does not hold. Thus, the result of Theorem 13 in  fails in providing an error bound for this LCP.

Secondly, if F is γ-strongly G-monotone and G is strongly δ-nonexpanding, then it is easy to verify that (35)F(x)-F(y),G(x)-G(y)c5G(x)-G(y)1+γc5α1+γx-y(1+γ)δ,x,yRn, where c5>0 is constant. Moreover, the conditions that both F and G are Hölder continuous (or both F and G are Lipschitz continuous) in Theorem 8 in this paper are removed. Thus, Theorem 8 is stronger than Theorem 2.5 in . Furthermore, by Theorem 2.1 in , the GNCP can be reformulated as general variational inequalities problem, and the conditions in Theorem 8 are also weaker than those in Theorem 3.1 in , Theorem 2 in , Theorem 3.1 in , and Theorem 3.1 in , respectively.

In the end of this paper, we will consider a special case of GNCP which was discussed in .

When 𝒦=R+m, the (2) reduces to the generalization of the classical nonlinear complementarity problem of finding vector x*Rn such that (36)F(x*)0,G(x*)0,F(x*)G(x*)=0. Combining this with Theorem 8, we can also immediately obtain the following conclusion.

Corollary 11.

Suppose that the hypotheses of Theorem 8 hold. Then there exists constant ρ2>0 such that (37)dist(x,X*)ρ2{[F(x)]-+[G(x)]-+[F(x)G(x)]1/2}2/(1+γ)δ,mmmmmmmmmmmmmmmmmmmmmmmmlllxRn.

Remark 12.

It is clear that if F is γ-uniform p-function and G is strongly δ-nonexpanding, for any x,yRn, then (38)max1im{[F(x)-F(y)]i[G(x)-G(y)]i}mmmllc1G(x)-G(y)1+γc1α1+γx-y(1+γ)δ, so the condition in Corollary 11 is largely extended than the condition that F is a uniform p-function with respect to G (i.e., γ=1,δ=1) in Theorem 3.3 in . Moreover, the conditions that both F and G are Lipschitz continuous in Theorem 3.3 in  are removed. Thus, Corollary 11 is stronger than Theorem 3.3 in .

3. Conclusion

In this paper, we established a global error bound on the generalized nonlinear complementarity problems over a polyhedral cone, which improves the result obtained for variational inequalities and the GNCP in [4, 5, 1113] by weakening the assumptions. Surely, we may use the error bound estimation to establish quick convergence rate of the methods for the GNCP under milder conditions. This is a topic for future research.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors wish to give their sincere thanks to the associated editor and two anonymous referees for their valuable suggestions and helpful comments which improve the presentation of the paper. This work was supported by the Natural Science Foundation of China (nos. 11171180, 11101303, 11171362, and 11271226), the Specialized Research Fund for the Doctoral Program of Chinese Higher Education (20113705110002, and 20120191110031), the Shandong Provincial Natural Science Foundation (ZR2010AL005), the Shandong Province Science and Technology Development Projects (2013GGA13034), the Domestic Visiting Scholar Project for the Outstanding Young Teacher of Shandong Province Universities (2013), and the Applied Mathematics Enhancement Program of Linyi University.

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