# An Improvement of Global Error Bound for the Generalized Nonlinear Complementarity Problem over a Polyhedral Cone

Authors: Hongchun Sun; Yiju Wang; Houchun Zhou; Shengjie Li
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

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.


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## Body

## 1. Introduction

Let𝒦={v∈Rm∣Av≥0,Bv=0} be a polyhedral cone in Rm for matrices A∈Rs×m, B∈Rt×m, and let 𝒦∘ be its dual cone; that is,
(1)𝒦∘={u∈Rm∣u=A⊤λ1+B⊤λ2,λ1∈R+s,λ2∈Rt}.
For continuous mappings, F,G:Rn→Rm, 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 [1]. The GNCP was deeply discussed [2–7] after the work in [8]. 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 [4], a global error bound estimation for GNCP with the mapping being γ-strongly monotone and Hölder continuous was established in [5], 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, 11–13].To end this section, we give some notations used in this paper. Vectors considered in this paper are taken in Euclidean spaceRn 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 1≤i≤n, respectively. For simplicity, we use (x;y) to denote vector (x⊤,y⊤)⊤, use I to denote the identity matrix with appropriate dimension, use x≥0 to denote a nonnegative vector x∈Rn, 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 mappingF:Rn→Rm is said to be γ-uniform p-function with respect to G:Rn→Rm if there are constants c1>0 and γ>0 such that
(3)max⁡1≤i≤m{[F(x)-F(y)]i[G(x)-G(y)]i}mmmmm≥c1∥G(x)-G(y)∥1+γ,mmmmmmmmmmml∀x,y∈Rn.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 [4], and if
(4)F(x)=Mx+p,G(x)=Nx+q
with M,N∈Rm×n, p,q∈Rm, then the above definition is equivalent in which the matrix M⊤N is a p-matrix [14]. For example, let
(5)M=(1-411),N=(1001),p=q=(00).
By Theorem 2.1.15 in [14], it is easy to verify that M⊤N is a p-matrix. However, letting x=(1;2), we note that x⊤M⊤Nx=-1<0 which shows that M⊤N 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 Definition1.Assumption 3.
For mappingsF,G and matrix A involved in the GNCP, we assume that(A1)
mappingF is γ-uniform p-function with respect to mapping G;(A2)
matrixA⊤ has full-column rank.In the following, we give the conclusion established in [2].Theorem 4.
A pointx*∈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*)=A⊤λ1*+B⊤λ2*.From Theorem4, under Assumption 3 (A2), similar to discussion in [4], 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⊤[(A⊤AL-1-I)B⊤]+[A⊤AL-1-I]+AL-1},V={A⊤{-AL-1B⊤[(A⊤AL-1-I)B⊤]+[A⊤AL-1-I]+AL-1}mmm+B⊤[(A⊤AL-1-I)B⊤]+[A⊤AL-1-I]-IA⊤{-AL-1B⊤[(A⊤AL-1-I)B⊤]+[A⊤AL-1-I]+AL-1}}.For the ease of description, letμ=F(x) and ν=G(x). Then, system (7) can be written as
(9)Aμ≥0,Bμ=0,μ⊤ν=0,Uν≥0,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.
LetS1:={ω∈R2m∣f(ω)=0}. Then, one has
(10)dist⁡(ω,S1)≤τ∥f(ω)∥1/2,
where τ>0 is a constant, ω=(μ;ν), and f(ω)=μ⊤ν.Proof.
For anyω∈R2m, let μ=(μ1,μ2,…,μm)⊤, ν=(ν1,ν2,…,νm)⊤, and
(11)ωi={μi1≤i≤m,νim+1≤i≤2m.
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 P∈R2m×2m such that ω=Pξ, and one has
(12)f(ω)=μ⊤ν=∑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+12m‍ξi2∑i=1m‍ξi2)1/2,ξ¯i={θξi1≤i≤m,ξim+1≤i≤2m.
It is easy to verify that0≤θ≤1 and
(14)g(ξ¯)=θ2∑i=1m‍ξi2-∑i=m+12m‍ξi2=(∑i=m+12m‍ξi2∑i=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¯∥≤∥P∥∥ξ-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=∥P∥1+θ(1-∑i=m+12m‍ξi2g(ξ)+∑i=m+12m‍ξi2)(∑i=1m‍ξi2)1/2=∥P∥g(ξ)(1+θ)(∑i=1m‍ξi2)(∑i=1m‍ξi2)1/2=∥P∥g(ξ)(1+θ)(∑i=1m‍ξi2)1/2=∥P∥g(ξ)(∑i=1m‍ξi2)1/2+θ(∑i=1m‍ξi2)1/2=∥P∥g(ξ)(∑i=1m‍ξi2)1/2+((∑i=m+12m‍ξi2)/(∑i=1m‍ξi2))1/2(∑i=1m‍ξi2)1/2=∥P∥g(ξ)(∑i=1m‍ξi2)1/2+(∑i=m+12m‍ξi2)1/2≤∥P∥g(ξ)(∑i=1m‍ξi2+∑i=m+12m‍ξi2)1/2≤∥P∥g(ξ)g(ξ)1/2=∥P∥g(ξ)1/2=∥P∥f(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,b∈R+,
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 [15] on the error bound for a polyhedral cone.Lemma 6.
For polyhedral coneP={x∈Rn∣D1x=d1,B1x≤b1} with D1∈Rl×n, B1∈Rm×n, d1∈Rl, and b1∈Rm, there exists a constant c2>0 such that
(18)dist⁡(x,P)≤c2[∥D1x-d1∥+∥(B1x-b1)+∥],∀x∈Rn.Before proceeding, we present the following definition introduced in [16] with constant δ=1.Definition 7.
The mappingG:Rn→Rm is said to be δ-strongly nonexpanding with a constant α>0, if ∥G(x)-G(y)∥≥α∥x-y∥δ, where δ>0.Now, we are at the position to state our main results in this paper.Theorem 8.
Suppose thatF 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,∀x∈Rn.Proof.
Using Lemma5, for any ω∈R2m, there exists ω¯∈S1 such that
(20)∥ω-ω¯∥≤τ∥[f(ω)]∥1/2,
where S1 is defined in Lemma 5. Let
(21)Ω={ω∈R2m∣A(I,0)ω≥0,B(I,0)ω=0,U(0,I)ω≥0,V(0,I)ω=0R2m∣A(I,0)ω}.
From (9), we have Ω*=Ω⋂‍S1. 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 ω∈S1, 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}=c3∥Ψ(ω)∥1≤c32s+2t+2m∥Ψ(ω)∥,
where c3 is a positive constant and the third and fourth inequalities follow from the fact that ∥x∥≤∥x∥1≤n∥x∥, for all x∈Rn.
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¯∥+2∥B(I,0)ω-B(I,0)ω¯∥mmm+∥U(0,I)ω-U(0,I)ω¯∥+2∥V(0,I)ω-V(0,I)ω¯∥≤(∥A(I,0)∥+2∥B(I,0)∥+∥U(0,I)∥mmm+2∥V(0,I)∥)∥ω-ω¯∥,
where the second equality follows from the fact that
(25)min⁡{a,b}=a-PR+(a-b),∀a,b∈R,
and the first inequality is by nonexpanding property of projection operator. Combining (24), one has
(26)∥Ψ(ω¯)∥≤∥Ψ(ω)∥+(∥A(I,0)∥+2∥B(I,0)∥mml+∥U(0,I)∥+2∥V(0,I)∥)∥ω-ω¯∥.
Combining (23) with (26), for any ω∈R2m, we have
(27)∥ω-ω*∥≤∥ω-ω¯∥+∥ω¯-ω*∥≤∥ω-ω¯∥+σ∥Ψ(ω¯)∥≤∥ω-ω¯∥+σ(∥Ψ(ω)∥+(∥A(I,0)∥+2∥B(I,0)∥mmmmmmmmmm+∥U(0,I)∥mmmmmmmlllllll+2∥V(0,I)∥)∥ω-ω¯∥)≤σ∥Ψ(ω)∥+[σ(∥A(I,0)∥+2∥B(I,0)∥mmmm+∥U(0,I)∥+2∥V(0,I)∥)+1]∥ω-ω¯∥≤σ∥Ψ(ω)∥+[σ(∥A(I,0)∥+2∥B(I,0)∥+∥U(0,I)∥mmmm+2∥V(0,I)∥)+1]τ∥[f(ω)]∥1/2≤η(∥Ψ(ω)∥+∥[f(ω)]∥1/2)≤η(∥Ψ(ω)∥1+∥[f(ω)]∥1/2)≤η(+∥V(0,I)ω∥1+∥[f(ω)]∥1/2∥(-A(I,0)ω)+∥1+∥(-U(0,I)ω)+∥1mmmm+∥B(I,0)ω∥1mmmm+∥V(0,I)ω∥1+∥[f(ω)]∥1/2)≤η(+m∥V(0,I)ω∥+∥[f(ω)]∥1/2s∥(-A(I,0)ω)+∥+s∥(-U(0,I)ω)+∥mmmm+t∥B(I,0)ω∥mmmm+m∥V(0,I)ω∥+∥[f(ω)]∥1/2)≤c4(∥[f(ω)]∥1/2∥(-A(I,0)ω)+∥+∥(-U(0,I)ω)+∥mmmm+∥B(I,0)ω∥mmmm+∥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)∥+2∥B(I,0)∥mmmmml+∥U(0,I)∥+2∥V(0,I)∥)+1]τ},
the seventh and ninth inequalities follow from the fact that
(29)∥x∥≤∥x∥1≤n∥x∥,∀x∈Rn,
and the last inequality follows by letting c4=ηmax⁡{s,t,m,1}.
For anyx∈Rn, 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)δ≤1α1+γ∥G(x)-G(x*)∥1+γ≤1c1α1+γmax⁡1≤i≤n{[F(x)-F(y)]i[G(x)-G(y)]i}≤1c1α1+γ∥F(x)-F(x*)∥∥G(x)-G(x*)∥≤12c1α1+γ{∥F(x)-F(x*)∥2+∥G(x)-G(x*)∥2}=12c1α1+γ∥ω-ω*∥2≤12c1α1+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)ω∥mmmmmmmmmm+∥V(0,I)ω∥+∥[f(ω)]∥1/2)2≤12c1α1+γc42(+∥V(0,I)ω∥+∥[f(ω)]∥1/2∥(A(I,0)ω)-∥+∥(U(0,I)ω)-∥mmmmmmmmmm+∥B(I,0)ω∥mmmmmmmmmm+∥V(0,I)ω∥+∥[f(ω)]∥1/2)2≤12c1α1+γc42{∥μ⊤ν∥1/2∥[Aμ]-∥+∥Bμ∥+∥[Uν]-∥+∥Vν∥mmmmmmmmmm+∥μ⊤ν∥1/2}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+b2≥2ab, for all a,b∈R, 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 Definition1, 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 [4]. In addition, the result in Theorem 8 is stronger than that in Theorem 13 in [4]. Thus, Theorem 8 is stronger than Theorem 13 in [4].
In the following, we also present an example to compare the condition in Theorem8 in this paper and that in Theorem 13 in [4].Example 10.
When𝒦=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 LCPX*={0}. In fact,
(32)X*={x∈R2∥x≥0,Mx≥0,x⊤Mx=0}={3+52(x1,x2)∈R2∣x1≥0,x2≥0,x1≥3x2,mmlx1=3-52x2orx1=3+52x2}={0}.
Clearly, M is a P matrix [17]FV. Thus, there exists constant τ>0 such that
(33)max⁡1≤i≤2{(x-y)i(Mx-My)i}≥τ∥x-y∥2.
For any x(ϵ):=(-ϵ;ϵ), ϵ≥0. 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)-0∥φ1(x(ϵ))=2ϵ4ϵ+ϵ+5ϵ⟶25+5
as ϵ→0. Thus, Theorem 8 provides a global error bound for this LCP.However, lettingx=(1;1), we note that x⊤Mx=-1<0 which shows that M is not positive definite, so the condition that F is strongly monotone in Theorem 13 in [4] does not hold. Thus, the result of Theorem 13 in [4] fails in providing an error bound for this LCP.Secondly, ifF 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)〉≥c5∥G(x)-G(y)∥1+γ≥c5α1+γ∥x-y∥(1+γ)δ,∀x,y∈Rn,
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 [5]. Furthermore, by Theorem 2.1 in [5], 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 [11], Theorem 2 in [12], Theorem 3.1 in [13], and Theorem 3.1 in [16], respectively.In the end of this paper, we will consider a special case of GNCP which was discussed in [11].When𝒦=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 Theorem8 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,mmmmmmmmmmmmmmmmmmmmmmmmlll∀x∈Rn.Remark 12.
It is clear that ifF is γ-uniform p-function and G is strongly δ-nonexpanding, for any x,y∈Rn, then
(38)max⁡1≤i≤m{[F(x)-F(y)]i[G(x)-G(y)]i}mmmll≥c1∥G(x)-G(y)∥1+γ≥c1α1+γ∥x-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 [11]. Moreover, the conditions that both F and G are Lipschitz continuous in Theorem 3.3 in [11] are removed. Thus, Corollary 11 is stronger than Theorem 3.3 in [11].


## 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, 11–13] 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.



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