# A Smoothing Inexact Newton Method for Generalized Nonlinear Complementarity Problem

Authors: Meixia Li; Haitao Che
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

Based on the smoothing function of penalized Fischer-Burmeister NCP-function, we propose a new smoothing inexact Newton algorithm with non-monotone line search for solving the generalized nonlinear complementarity problem. We view the smoothing parameter as an independent variable. Under suitable conditions, we show that any accumulation point of the generated sequence is a solution of the generalized nonlinear complementarity problem. We also establish the local superlinear (quadratic) convergence of the proposed algorithm under the BD-regular assumption. Preliminary numerical experiments indicate the feasibility and efficiency of the proposed algorithm.


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## Body

## 1. Introduction

Consider the generalized nonlinear complementarity problem, denoted by GNCP (F,G,𝒦), which is to find a vector x∈Rn such that(1.1)F(x)∈K,G(x)∈K∘,F(x)⊤G(x)=0,
where F,G:Rn→Rm are continuously differentiable mappings. 𝒦 is a nonempty closed convex cone in Rm and 𝒦∘ denotes its polar cone.GNCP (F,G,𝒦) finds important applications in many fields, such as engineering and economics, and is a wide class of problems that contains the classical nonlinear complementarity problem (abbreviated as NCP); see [1–3] and references therein. To solve it, one usually reformulates it as a minimization problem over a simple set or an unconstrained optimization problem; see [4] for the case that K is a general cone, and see [3, 5] for the case that 𝒦=Rn+. The conditions under which a stationary point of the reformulated optimization is a solution of the GNCP (F,G,𝒦) were also provided in the literature.In this paper, we consider the GNCP (F,G,𝒦) for the case that m=n, and 𝒦 is a polyhedral cone in Rn; that is, there exist A∈Rs×n and B∈Rt×n such that(1.2)K={v∈Rn∣Av≥0,Bv=0}.
It is easy to verify that its polar cone 𝒦∘ has the following representation:(1.3)K∘={u∈Rn∣u=A⊤λ+B⊤μ,λ∈Rs,λ≥0,μ∈Rt}.In the following, the GNCP (F,G,𝒦) is specialized over a polyhedral cone, and in the subsequent analysis, we abbreviate it as GNCP for simplicity. In [1], Andreani et al. reformulated the problem as a smooth optimization problem with simple constraints and presented the sufficient conditions under which a stationary point of the optimization problem is a solution of the concerned problem. Wang et al. [6] reformulated the problem as a system of nonlinear and nonsmooth equations and proposed a nonsmooth L-M method to solve this problem and proved that the algorithm is both globally and superlinearly convergent under mild assumptions. Zhang et al. [7] rearranged the GNCP over a polyhedral as a smoothing system of equations, then developed a smoothing Newton-type method for solving it, and proved that their method has local superlinear (quadratic) convergence under certain conditions. There are a lot of computations to decide whether the linear system is solvable or not, which is in Step 2 of the algorithm presented in [7]. The inexact approach is one way to overcome this difficulty. Inexact Newton methods have been proposed for solving NCP [8–10]. Their main idea is to solve the linear system only approximately. It seems reasonable to ask if this kind of method can be applied to the GNCP, and this actually constitutes the main motivation of this paper. In this paper, we propose a new smoothing inexact Newton algorithm with non-monotone line search for solving GNCP by using a new type of smoothing function. We view the smoothing parameter as an independent variable. The forcing parameter of inexact Newton method links the norm of residual vector to the norm of mapping at the current iterate. Under suitable conditions, we show that any accumulation point of the generated sequence is a solution of the GNCP, and we also establish the local superlinear (quadratic) convergence of the proposed algorithm under the BD-regular assumption. Some numerical examples indicate the feasibility and efficiency of the proposed algorithm.The rest of this paper is organized as follows. In Section2, we show some preliminaries. Stationary point and nonsingularity conditions of the GNCP are discussed in Section 3. In Section 4, we propose a kind of algorithm for solving GNCP and obtain the convergence properties of this kind of algorithm. Numerical experiments are exhibited in Section 5 and the conclusion is stated in the last section.At the end of this section, we indicate some standard notions used in this paper. For a continuously differentiable functionF:Rn→Rm, we denote the Jacobian of F at x∈Rn by F′(x)∈Rm×n, whereas the transposed Jacobian is denoted by ∇F(x). In particular, if m=1, ∇F(x) is a column vector. We use x⊤y to denote the inner product of vectors x,y∈Rn, and use [x]i or xi to denote the ith component of the vector x∈Rn. The null space of a matrix B is denoted by 𝒩(B).


## 2. Preliminaries

In this section, we first review some definitions and basic results.Definition 2.1.
A matrixM∈Rn×n is said to be a P0-matrix if every principle minor of M is nonnegative.Definition 2.2.
A functionF:Rn→Rn is said to be a P0-function if for all x,y∈Rn with x≠y, there exists an index i0∈{1,2,…,n} such that
(2.1)xi0≠yi0,(xi0-yi0)[Fi0(x)-Fi0(y)]≥0.
For vectora∈Rn, we use Da for diag(a). For a P0-matrix, the following conclusion holds [11].Lemma 2.3.
IfM∈Rn×n is a P0-matrix, then every matrix of the form
(2.2)Da+DbM
is nonsingular for all positive definite diagonal matrices Da,Db∈Rn×n.Definition 2.4.
Suppose thatG:Rn→Rm is locally Lipschitz function. G is said to be semismooth at x if G is directionally differentiable at x and
(2.3)lim⁡V∈∂G(x+th′),h′→h,t→0+{Vh′}
exist for any h∈Rn, where ∂G(x) denotes the generalized derivative in [12].Definition 2.5.
Suppose thatG:Rn→Rm is locally Lipschitz function. G is said to be strongly semismooth at x∈Rn if G(x) is semismooth at x and for any V∈∂G(x+h) and h→0, it holds that
(2.4)G(x+h)-G(x)-Vh=O(‖h‖2).
The concept of semismoothness was originally introduced by Mifflin for functionals [13]. Qi and Sun extended the definition of semismooth function to vector-valued functions [14]. Convex functions, smooth functions, and piecewise linear functions are examples of semismooth functions. A function is semismooth at x if and only if all its component functions are semismooth. The composition of semismooth functions is still a semismooth function.Lemma 2.6 (see [14]).
Suppose thatφ:Rn→Rm is a locally Lipschitz function and semismooth at x. Then(a)
for anyV∈∂φ(x+h),h→0,(2.5)Vh-φ′(x;h)=o(‖h‖),(b)
for anyh→0,(2.6)φ(x+h)-φ(x)-φ′(x;h)=o(‖h‖).In [6], Wang et al. reformulated the GNCP as a system of nonlinear equations based on the following Fischer function [15] ϕ(a,b)=a2+b2-a-b for a,b∈R. In [7], Zhang et al. established a type of smoothing reformulation of GNCP based on the following smoothing approximation function to the Fischer function ϕ(ϵ,a,b)=a2+b2+αϵ2-a-b for a,b,ϵ∈R, where α>0 is a constant.In this paper, we use the following smoothing approximation function:(2.7)ϕα(ϵ,a,b)=a+b-a2+b2+2ϵ2+α4(a+a2+4ϵ2)(b+b2+4ϵ2)
of the penalized Fischer-Burmeister NCP-function:(2.8)ϕα(a,b)=a+b-a2+b2+αa+b+,
where α>0,a,b,ϵ∈R, and t+=max{0,t}. This NCP-function has turned out to have stronger theoretical properties than the widely used Fischer-Burmeister function and other NCP-functions suggested previously (see [16]). The latter term penalizes violations of the complementarity condition and plays a significant role from both a theoretical and a practical point of view.It is easy to see that the following Lemma is true from [6].Lemma 2.7.
x∈Rn is a solution of the GNCP if and only if there exist λ∈Rs and μ∈Rt, such that
(2.9)Φα(AF(x),λ)=0,BF(x)=0,G(x)-A⊤λ-B⊤μ=0,
where Φα(a,b)=(ϕα(a1,b1),ϕα(a2,b2),…,ϕα(as,bs))⊤ for a,b∈Rs.Based on the relation betweenϕα(·,·) and ϕα(·,·,·), we can establish the following smoothing function to the GNCP.Denote(2.10)Φα(ϵ,AF(x),λ)=(ϕα(ϵ,[AF(x)]1,[λ]1)ϕα(ϵ,[AF(x)]2,[λ]2)⋮ϕα(ϵ,[AF(x)]s,[λ]s)),ψα(ϵ,x,λ,μ)=(Φα(ϵ,AF(x),λ)BF(x)G(x)-A⊤λ-B⊤μ),Hα(ϵ,x,λ,μ)=(ϵψα(ϵ,x,λ,μ)).For simplicity, we lety=(x,λ,μ) and z=(ϵ,y) and denote(2.11)fα(z)=12‖ψα(z)‖2,Tα(z)=12‖Hα(z)‖2=12ϵ2+fα(z).From Lemma2.7, we can see that x* is a solution of the GNCP if and only if there exist λ*∈Rs,μ*∈Rt such that z*=(0,x*,λ*,μ*) is a global minimizer with zero objective function value of the unconstrained optimization problem:(2.12)min⁡z∈R1+n+s+tTα(z).By simple calculation, we can see that the following lemma is true.Lemma 2.8.
(1) The functionΦα(ϵ,AF(x),λ) is continuously differentiable for ϵ>0 and it holds that
(2.13)∂Φα(ϵ,AF(x),λ)⊆(c,[D1(x)+D3(x)]AF′(x),[D2(x)+D4(x)]),
where(2.14)c=vec⁡{ci:i∈N},N={1,2,…,n},ci=-2ϵ[AF(x)]i2+[λ]i2+2ϵ2+α4[([AF(x)]i+[AF(x)]i2+4ϵ2)4ϵ[λ]i2+4ϵ2+([λ]i+[λ]i2+4ϵ2)4ϵ[AF(x)]i2+4ϵ2],D1(x)=diag⁡{1-[AF(x)]i[AF(x)]i2+[λ]i2+2ϵ2:i∈N},D2(x)=diag⁡{1-[λ]i[AF(x)]i2+[λ]i2+2ϵ2:i∈N},D3(x)=diag⁡{α4(1+[AF(x)]i[AF(x)]i2+4ϵ2)([λ]i+[λ]i2+4ϵ2):i∈N},D4(x)=diag⁡{α4(1+[λ]i[λ]i2+4ϵ2)([AF(x)]i+[AF(x)]i2+4ϵ2):i∈N}.
(2)Hα(z) is semismooth on R1+n+s+t and is strongly semismooth on R1+n+s+t if F′(x) and G′(x) are both Lipschitz continuous on Rn.
(3)Tα(z) is continuously differentiable on R1+n+s+t with ∇Tα(z)=V⊤Hα(z) for any V∈∂Hα(z) and fα(ϵ,y) is continuously differentiable with ∇fα(0,y)=V⊤ψα(0,y) for any V∈∂ψα(0,y).


## 3. Stationary Point and Nonsingularity Conditions

Generally, for an optimization problem, we can obtain its stationary point when we use the existing optimization methods to solve it. Now we should study how to guarantee that every stationary point of (2.12) is a solution of the GNCP. In the following, we will discuss the conditions.Theorem 3.1.
Letz=(ϵ,x,λ,μ) be a stationary point of (2.12). If ϵ≥0 and ∇F(x)-1∇G(x) is positive definite in 𝒩(B), then x is a solution of the GNCP.Proof.
Define(3.1)U=Φα(ϵ,AF(x),λ),V=BF(x),W=G(x)-A⊤λ-B⊤μ.
Since z=(ϵ,x,λ,μ) is a stationary point of (2.12), one has
(3.2)∇Tα(z)=0.
From Lemma 2.8, we have
(3.3)(1c⊤000∇F(x)A⊤[D1(x)+D3(x)]∇F(x)B⊤∇G(x)0[D2(x)+D4(x)]0-A000-B)(ϵUVW)=0,
that is,
(3.4)ϵ+c⊤U=0,∇F(x)A⊤[D1(x)+D3(x)]U+∇F(x)B⊤V+∇G(x)W=0,(D2(x)+D4(x))U-AW=0,-BW=0.
From the forth equation of (3.4), we have W∈𝒩(B). Premultiplying the second equation of (3.4) by W⊤∇F(x)-1, one has
(3.5)W⊤A⊤[D1(x)+D3(x)]U+W⊤B⊤V+W⊤∇F(x)-1∇G(x)W=0.
Combining (3.5) with the third and forth equations of (3.4), we obtain
(3.6)U⊤[D2(x)+D4(x)][D1(x)+D3(x)]U+W⊤∇F(x)-1∇G(x)W=0.
From Lemma2.8, we have that [D2(x)+D4(x)][D1(x)+D3(x)] is positive definite. Since W∈𝒩(B) from the forth equation of (3.4), together with the positive definiteness of ∇F(x)-1∇G(x) in 𝒩(B), we have
(3.7)U=0,W=0.
SubtitutingU=0 into the first equation of (3.4), we have
(3.8)ϵ=0.
Together the second equation of (3.4) with (3.8) and (3.7), we have
(3.9)∇F(x)B⊤V=0.
Since∇f(x) is nonsingular, premultiplying (3.9) by F(x)⊤∇F(x)-1, one has
(3.10)‖V‖2=0.
Hence, V=0. We complete the proof.Theorem 3.2.
Letz=(ϵ,x,λ,μ) be a stationary point of (2.12), ϵ>0,B has full row rank, and ∇F(x)∇G(x)⊤ is positive definite, then V is nonsingular for any V∈∂Hα(z).Proof.
By Lemma2.8, we know that any element V∈∂Hα(z) can be written as
(3.11)V⊤=(1c⊤000∇F(x)A⊤[D1(x)+D3(x)]∇F(x)B⊤∇G(x)0[D2(x)+D4(x)]0-A000-B),
where D1(x),D2(x),D3(x), and D4(x) are defined in Lemma 2.8.
In order to complete the proof, we only need to prove that(3.12)D=(∇F(x)A⊤[D1(x)+D3(x)]∇F(x)B⊤∇G(x)D2(x)+D4(x)0-A00-B)
is nonsingular.
That is to say, if and only if(3.13)D⊤=([D1(x)+D3(x)]A∇F(x)⊤D2(x)+D4(x)0B∇F(x)⊤00∇G(x)⊤-A⊤-B⊤)
is nonsingular.
Suppose that(3.14)([D1(x)+D3(x)]A∇F(x)⊤D2(x)+D4(x)0B∇F(x)⊤00∇G(x)⊤-A⊤-B⊤)(pqr)=0,
which means that
(3.15)[D1(x)+D3(x)]A∇F(x)⊤p+[D2(x)+D4(x)]q=0,B∇F(x)⊤p=0,∇G(x)⊤p-A⊤q-B⊤r=0.
According to the first equation of (3.15), we have
(3.16)q=-[D2(x)+D4(x)]-1[D1(x)+D3(x)]A∇F(x)⊤p.
Combining (3.16) with the third equation of (3.15), one has
(3.17){∇G(x)⊤+A⊤[D2(x)+D4(x)]-1[D1(x)+D3(x)]A∇F(x)⊤}p-b⊤r=0.
Premultiplying by p⊤∇F(x) and recalling the second equation of (3.15), we get
(3.18)p⊤{∇F(x)∇G(x)⊤+(A∇F(x)⊤)⊤[D2(x)+D4(x)]-1[D1(x)+D3(x)]A∇F(x)⊤}p=0.
By using the assumption that ∇F(x)∇G(x)⊤ is positive, we obtain
(3.19)p=0,
which, combining with (3.16), gives
(3.20)q=0.
By using the assumption that B has full row rank and the third equation of (3.15), we obtain
(3.21)r=0,
which completes the proof.


## 4. Algorithm and Convergence Property

In this section, we formally present our smoothing inexact Newton-type algorithm with nonmontone line search for solvingHα(z)=0 by using the smoothing penalized FB function ϕα(ϵ,a,b). This nonmontone line search method was used to solve the NCP problem in [17]. Furthermore, we show the local superlinear (quadratic) convergence properties of the algorithm.Algorithm 4.1.
Step 1.
Take constantsδ,γ∈(0,1),α>0, and ϵ0>0 such that γϵ0<1. Choose σ∈(0,1/2).
Letϵ¯=(ϵ0,0)∈R++×Rn+s+t and y0=(x0,λ0,μ0)∈Rn+s+t be an arbitary point.
Letz0=(ϵ0,y0),C0=Tα(z0), and Q0=1. Let ηmin and ηmax be two constants such that 0≤ηmin<ηmax<1. Choose η0∈[ηmin,ηmax]. Set ρ(z0)=γmin{1,Tα(z0)} and k:=0.Step 2.
IfHα(zk)=0, then stop; otherwise, let ρk=ρ(zk).Step 3.
ComputeΔzk=(Δϵk,Δxk,Δλk,Δμk)∈R1+n+s+t by
(4.1)Hα(zk)+Hα′(zk)Δzk=ρkϵ¯.Step 4.
Letξk=δm, where m is the smallest nonnegative integer such that
(4.2)Tα(zk+δmΔzk)≤[1-2σ(1-γϵ0δm)]Ck.Step 5.
Setzk+1=zk+ξkΔzk, k=k+1, and
(4.3)ρ(zk)=min⁡{γ,γTα(zk),ρ(zk-1)}.Step 6.
Chooseηk∈[ηmin,ηmax]; set
(4.4)Qk=ηk-1Qk-1+1,Ck=ηk-1Qk-1Ck-1+Tα(zk)Qk.
Go to Step2.
From Algorithm4.1 and [17], it is easy to see that the following remark is true.Remark 4.2.
Let the sequence{Ck} and {zk=(ϵk,xk,λk,μk)} be generated by Algorithm 4.1.(i)
Ck+1≤Ck for any k.(ii)
Tα(zk)≤Ck for any k.(iii)
ρ(zk+1)≤ρ(zk) for any k.(iv)
ϵ0ρ(zk)≤ϵk for any k.(v)
ϵk>0 and ϵk+1≤ϵk for any k.Theorem 4.3.
Suppose that{zk=(ϵk,xk,λk,μk)} is a sequence generated by Algorithm 4.1, B has full row rank, and ∇F(xk)∇G(xk)⊤ is positive definite, ϵk>0. Then Algorithm 4.1 is well defined.Proof.
From Theorem3.2, we know that Hα′(zk) is nonsingular. Hence, Step 3 is well defined at the kth iteration.
In the following, we show that the Step4 is well defined. For ξ∈(0,1), we let
(4.5)Rk(ξ)=Tα(zk+ξΔzk)-Tα(zk)-ξTα′(zk)Δzk,
and then
(4.6)Tα(zk+ξΔzk)=Rk(ξ)+Tα(zk)+ξTα′(zk)Δzk=Rk(ξ)+Tα(zk)+2ξHα(zk)⊤Hα′(zk)Δzk=Rk(ξ)+Tα(zk)+2ξHα(zk)⊤(-Hα(zk)+ρkϵ¯)=Rk(ξ)+Tα(zk)-2ξHα(zk)⊤Hα(zk)+2ξHα(zk)⊤ρkϵ¯≤Rk(ξ)+Tα(zk)-2ξHα(zk)⊤Hα(zk)+2ξρk‖Hα(zk)‖ϵ0=Rk(ξ)+Tα(zk)+2ξ(-Tα(zk)+ϵ0ρk‖Hα(zk)‖).
IfTα(zk)≤1, then ∥Hα(zk)∥≤1, which, together with (4.3), implies that
(4.7)ϵ0ρk‖Hα(zk)‖≤ϵ0γTα(zk).
If Tα(zk)>1, then ∥Hα(zk)∥≤Tα(zk). Combining with (4.3), we have
(4.8)ϵ0ρk‖Hα(zk)‖≤ϵ0γTα(zk).
As a result, for any k,
(4.9)ϵ0ρk‖Hα(zk)‖≤ϵ0γTα(zk).
From (4.6) and (4.9), one has
(4.10)Tα(zk+ξΔzk)≤Rk(ξ)+Tα(zk)+2ξ(-Tα(zk)+ϵ0γTα(zk))≤|Rk(ξ)|+(1-2ξ(1-ϵ0γ))Tα(zk).
According to Lemma2.8, we know that Tα(z) is continuously differentiable. Hence, there exists ξ¯>0 such that for any ξ∈(0,ξ¯),1-2ξ(1-ϵ0γ)>0 and
(4.11)Tα(zk+ξΔzk)≤(1-2ξ(1-ϵ0γ))Tα(zk)≤(1-2ξ(1-ϵ0γ))Ck,
which completes the proof.Theorem 4.4.
Suppose that{zk=(ϵk,xk,λk,μk)} is an infinite sequence generated by Algorithm 4.1. Then the arbitrary accumulation point ẑ of the sequence {zk} is a stationary point of (2.12).Proof.
(i) Suppose thatẑ=(ϵ̂,ŷ) is an arbitrary accumulation point of {zk}, then there exists an infinite subsequence K1⊆{1,2,…} such that {zk}K1→ẑ as k∈K1 and k→∞. Without loss of generality, we assume that K1={1,2,…}. That is,
(4.12)lim⁡k→∞zk=ẑ.
According to Remark 4.2 (i), (iii), and (v), we obtain that the sequence {Ck},{ρk}, and {ϵk} are convergent. Without loss of generality, we assume that
(4.13)lim⁡k→∞Ck=Ĉ,lim⁡k→∞ρk=ρ̂,lim⁡k→∞ϵk=ϵ̂.
It is easy to see thatĈ≥0,ρ̂≥0, and ϵ̂≥0. By Remark 4.2 (i) and (ii), we know that
(4.14)0≤Tα(zk+1)≤Ck+1≤Ck≤C0,
which means that {Tα(zk)} is bounded.
In the following, suppose thatρ̂≠0. Without loss of generality, we assume that {Tα(zk)} is convergent and denote
(4.15)T̂=lim⁡k→∞Tα(zk).
Combining (4.3) with the assumption ρ̂≠0, we have T̂>0. Furthermore, by using (4.14) and Remark 4.2 (iv), we obtain that Ĉ>0 and ϵ̂>0. Hence, we can deduce that
(4.16)T̂=Hα(ẑ)⊤Hα(ẑ),T̂≤Ĉ,(4.17)ρ̂={γ,ifT̂>1,γT̂,ifT̂≤1.
We now break up the proof of (i) into two cases.
(1) Assume thatαk≥e>0 for all k, where e is a constant. In this case, by (4.4) and (4.2), it follows that
(4.18)Ck+1=ηkQKCk+Tα(zk)Qk+1≤ηkQKCk+Ck-2σ(1-γϵ0)αkCkQk+1=Ck-2σ(1-γϵ0)αkQk+1Ck≤Ck-2σ(1-γϵ0)eQk+1Ck,
for any k, which, together with the boundness of {Ck}, yields
(4.19)∑n=1∞2σ(1-γϵ0)eQk+1Ck<∞.
On the other hand, byηmax∈[0,1) and the definition of Qk given in (4.4), we have
(4.20)Qk+1=1+∑i=0k∏j=0iηk-j≤1+∑i=0kηmax⁡i+1≤∑i=0∞ηmax⁡i=11-ηmax⁡,
for any k.
As a result, we obtain thatlimk→∞Ck=0, which is a contradiction with Ĉ>0.
(2) Assume thatlimk→∞αk=0. The stepsize αk̂=αk/δ does not satisfy the line search condition (4.2) for any sufficiently large k, that is,
(4.21)Tα(zk+αk̂Δzk)>[1-2σ(1-γϵ0)αk̂]Ck
holds for sufficiently large k. Since Tα(zk)≤Ck, the aforementioned inequality becomes
(4.22)Tα(zk+αk̂Δzk)-Tα(zk)αk̂>2σ(1-γϵ0)Ck.
Sinceϵ̂>0 and Tα(·) is continuously differentiable at ẑ, from Lemma 2.8, we have
(4.23)-2σ(1-γϵ0)T̂=-2σ(1-γϵ0)Ĉ≤Tα′(ẑ)Δẑ=2Hα(ẑ)⊤(-Hα(ẑ)+ρ̂ϵ¯)≤2(-T̂+ϵ0ρ̂‖Hα(ẑ)‖)≤2(-T̂+ϵ0γT̂)=2(-1+ϵ0γ)T̂,
where the first equality holds from (4.4) (in the form of limit), the first inequality holds from (4.22) (in the form of limit); the second equality holds from (4.1) (in the form of limit), and the third inequality holds from ρ̂∥Hα(ẑ)∥≤γT̂ by using (4.17). Hence it follows from (4.23) and T̂>0 that -1+γϵ0≥-σ(1-γϵ0), which contradicts the fact that σ∈(0,1/2) and γϵ0<1. We complete the proof.Similar to the proof of Theorem  4.3 in [7], the following theorem holds.Theorem 4.5.
Suppose that{zk=(ϵk,xk,λk,μk)} is an infinite sequence generated by Algorithm 4.1, ẑ is the arbitrary accumulation point of the sequence {zk}, and ẑ is a BD-regular solution of H(z)=0. Then(1)
the pointx̂ is a solution of the GNCP;(2)
the sequence{zk} converges to ẑ superlinearly. In particular, if F′ and G′ are locally Lipschitz continuous at ẑ, then {zk} converges to ẑ Q-quadratically.


## 5. Numerical Experiments

In this section, we implement Algorithm4.1 for solving some GNCPs in order to see the behavior of Algorithm 4.1. The parameters used in the algorithm are chosen as follows:(5.1)α=0.01,ϵ0=2.25,σ=0.29,γ=0.02,δ=0.2.The following problems are tested in [5–7].Example 5.1.
Consider the implicit complementarity problems with the following form: findy∈Rn such that
(5.2)y-m(y)≥0,F(y)≥0,F(y)T(y-m(y))=0,
where mi:Rn→R,i=1,2,…,n, and
(5.3)F(y)=Ay+b=(2-100⋯00-12-10⋯000-12-1⋯00⋯⋯0000⋯2-10000⋯-12)y+(111⋮11),
and m(y)=ψ(Ay+b) with ψ:Rn→Rn being twice continuously differentiable. The following choices of function ψ define our test problems.(1)
ψi(x)=-0.5-xi,i=1,2,…,n,(2)
ψi(x)=-1.5xi+0.25xi2,i=1,2,…,n.The numerical results are shown in Tables1, 2, and 3 with the following four kinds of initial points:Table 1
Numerical results of Example5.1 with n=4.
ψSTITTVCPU(1)(a)99.7977×10-130.0163(2)(a)94.7750×10-140.0182(1)(b)123.9079×10-130.0190(2)(b)132.6126×10-130.0191(1)(c)156.9153×10-130.0210(2)(c)196.2238×10-130.0224(1)(d)131.0872×10-130.0186(2)(d)125.0859×10-130.0189Table 2
Numerical results of Example5.1 with n=8.
ψSTITTVCPU(1)(a)196.1362×10-130.1261(2)(a)155.5411×10-140.1142(1)(b)171.5028×10-130.1902(2)(b)172.4160×10-130.2184(1)(c)192.4432×10-130.2018(2)(c)222.6420×10-130.2215(1)(d)336.0168×10-130.2861(2)(d)273.2360×10-130.1839Table 3
Numerical results of Example5.1 with n=12.
ψSTITTVCPU(1)(a)242.7819×10-130.1453(2)(a)227.7064×10-140.1962(1)(b)182.0643×10-130.2003(2)(b)192.6558×10-130.2216(1)(c)212.9380×10-130.2421(2)(c)233.5009×10-130.2432(1)(d)459.6621×10-130.3056(2)(d)327.5216×10-130.2343(a)(0,0,…,0)⊤, (b)(-0.5,-0.5,…,-0.5)⊤,(c)(-1,-1,…,-1)⊤, (d)(0.5,0.5,…,0.5)⊤.In Tables1, 2, and 3, ST denotes initial point, IT is the iterative number, TV is the final value of T when the algorithm terminates, and CPU denotes the computing time in the computer, respectively. For the starting point, we choose λ0=(0.5,0.5,…,0.5)⊤∈Rn and the termination criterion for the Algorithm is ∥T(zk)∥≤10-6.From Tables1, 2, and 3, we can see that Algorithm 4.1 is efficient in solving this kind of problem.In the following, we will compare Algorithm4.1 (denoted by Inexact) with the exact Newton algorithm with nonmontone line search (denoted by Exact). Here, our test problem is the previous problem (1), that is,(5.4)ψi(x)=-0.5-xi,i=1,2,…,n.The initial point is(a) or (b). The numerical results are shown in Table 4.Table 4
Comparison of numerical results withn=800.
STAlgorithmITTVCPU(a)Inexact365.4366×10-135.4218(a)Exact879.6221×10-1321.3326(b)Inexact312.6359×10-134.1453(b)Exact796.4806×10-1319.1962From Table4, we can see that Algorithm 4.1 is prior to the exact smoothing Newton method when the GNCP is of relatively large scale.


## 6. Conclusion

In this paper, we combine the smoothing function of penalized Fischer-Burmeister NCP-function with nonmontone line search in [17] to present a new smoothing algorithm to solve the generalized nonlinear complementarity problem. We obtain that the iteration sequence generated by Algorithm 4.1 converges to a solution of the generalized nonlinear complementarity problem locally superlinear (quadratic). Preliminary numerical results show the efficiency of the algorithm.



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*Source: 401835-2012-04-12.xml*