An Interval Matrix Based Generalized Newton Method for Linear Complementarity Problems

The penalty equation of LCP is transformed into the absolute value equation, and then the existence of solutions for the penalty equation is proved by the regularity of the interval matrix. We propose a generalized Newton method for solving the linear complementarity problem with the regular interval matrix based on the nonlinear penalized equation. Further, we prove that this method is convergent. Numerical experiments are presented to show that the generalized Newton method is effective.


Introduction
The linear complementarity problem, denoted by ( ) , LCP M q , is to find a vector where n n M R × ∈ is a given matrix and n q R ∈ is a given vector.This problem serves as a unified formulation of linear and quadratic programming problems as well as of two-person (noncooperative) matrix-games, and has several important applications in economics and engineering sciences; see Cottle, Pang, and Stone [1] and its references.
There exist several methods for solving ( ) , LCP M q , such as projection method, multi splitting method, inte- rior point method, and the nonsmooth Newton method, smoothing Newton method, homotopy method etc. See [1]- [6] and its references.
In [7], it given a nonlinear penalized Equation (1.2) corresponding to linear complementarity problem (1.1).Find where 1 λ > is the penalized parameter, The nonlinear penalized problems (1.2) corresponding to the linear complementarity problem (1.1), which its research has achieved good results.In 1984, Glowinski [1] studied nonlinear penalized Equation (1.2) in n R , and proved the convergence of penalized equation that matrix A was symmetric positive definite.In 2006, Wang et al. [8] presented a power penalty function approach to the linear complementarity problem arising from pricing American options.It is shown that the solution to the penalized equation converges to that of the linear complementarity problem with matrix is positive definite.In 2008, Yang [7] proved that solution to this penalized Equations (1.2) converged to that of the LCP at an exponential rate for a positive definite matrix case where the diagonal entries were positive and off-diagonal entries were not greater than zero.The same year, Wang and Huang [9] presented a penalty method for solving a complementarity problem involving a secondorder nonlinear parabolic differential operator, and defined a nonlinear parabolic partial differential equation (PDE) approximating the variational inequality using a power penalty term with a penalty constant 1 λ > , a power parameter k >0 and a smoothing parameter ε .And prove that the solution to the penalized PDE converges to that of the variational inequality in an appropriate norm at an arbitrary exponential rate of the form ) . Under some assumptions, Li [10] [11] proved that the solution to this equation converges to that of the linear complementarity problem with A is a strict row diagonally dominant upper triangular P-matrix when the penalty parameter approaches to infinity and the convergence rate was also exponential.It is worth mentioning that the penalty technique has been widely used solving nonlinear programming, but it seems that there is a limited study for the LCP.
Although the studies solving for the linear complementarity problem based on the nonlinear penalized equation have good results.But there is no method that is given for solving the nonlinear penalized equation.Throughout the paper, we propose a generalized Newton method for solving the nonlinear penalized equation with under the suppose [ ] is regular.So the method can better to solve linear complementarity problem.We will show that the proposed method is convergent.Numerical experiments are also given to show the effectiveness of the proposed method.

Preliminaries
Some words about our notation: I refers to the identity matrix, and ⋅ represents the 2-norm.For The definition of interval matrix arises from the linear interval equations [12], given two matrices

{ }
, : Proof: By definition of ( ) , for every x, we have By the assumptions, we have ( ) is said to be a P-matrix if all its principal minors are positive.Lemma 3 [1]: A matrix ∈ is a P-matrix if and only if the ( ) , LCP q M has a unique solution for all vectors n q R ∈ .

Generalized Newton Method
In this section, we will propose that a new generalized Newton method based on the nonlinear penalized equation for solving the linear complementarity problem.Because when 1 k > , penalty term of the nonlinear penalized equation (1.2) is none Lipschitz continuously, hence we only discusses a case that 1 k = .So from nonlinear penalized equation (1.2), we have that

Mx
x q These case penalty problems for the continuous Variational Inequality and the linear complementarity problems are discussed in [2] [13].
Let us note Thus, nonlinear penalized equation (3.1) is equivalent to the equation ( ) A generalized Jacobian ( ) ( ) ( )( ) ) By [14], its equivalent to ( ) ( ) = has a unique solution if and only if the interval matrix [ ] + is P-matrix, which implies that the LCP has a unique solution for any , from the relation between the (3.1) and the LCP (3.5), we can easily deduce that the (3.1) is uniquely solvable for any Step 1: Choose an arbitrary initial point Step 2: for the k λ , computer Step 3: If x λ = go to step 4. Otherwise, 1 i i = + go to step 2.

The Convergence of the Algorithm
We will show that the sequence { } 1 x such that ( ) x λ is unbounded, Thus, there exists an infinite nonzero subsequence , and 0, is bounded.Hence, exists convergence subsequence and assume that convergence point is x  , and satisfy ( ) is regular, we know that ( ) is exists and hence 0 x =  , contradicting to the fact that 1 x =  .Consequently, the sequence x λ is bounded and there exists an accumulation point k Under a somewhat restrictive assumption we can establish finite termination of the generalized Newton iteration at a penalized equation solution as follows.Proof: Suppose that k x is a solution of nonlinear penalized equation.By the lemma 1, ) and by the lemma 2, we have ( ) Letting i → ∞ and taking limits in both sides of the last inequality above, we have Proof: Since M is P-Matrix, then the ( ) , LCP M q has a unique solution, let the solution denote * x , by the assumptions of the theorem, the generalized Newton iteration (3.6) linearly converges to a solution k x of the nonlinear penalized equation (3.1).
we have

Numerical Experiments
In this section, we give some numerical results in order to show the practical performance of Algorithm 2.1.Numerical results were obtained by using Matlab R2007 (b) on a 1G RAM, 1.86 Ghz Intel Core 2 processor.Throughout the computational experiments, the parameters were set as 1.0 8 e ε = − , 0 10 λ = , 2 µ = .

Example 1:
The matrix A of linear complementarity problem ( ) , LCP A b of as follows (This example appear in the Geiger and Kanzow [15], Jiang and Qi [16], YONG Long-quan, DENG Fang-an, CHEN Tao [17]): The computational results are shown in Table 1.This 0 x is initial point, k is number of inner iterations, the outer iteration number is m , * x is iteration results.Example 2: The matrix A of linear complementarity problem ( ) , LCP A b of as follows (This example appear in the Geiger and Kanzow [15], Jiang and Qi [16], YONG Long-quan, DENG Fang-an, CHEN Tao [17]): The computational results are shown in Table 2.This 0 x is initial point, k is number of inner iterations, the outer iteration number is m , * x is iteration results.
column vectors, Tx refers to the transpose of the x , component of x which is positive , negative or zero, respectively.

Lemma 1 :
Assume that interval matrix [ ] matrix whose diagonal entries are equal 1,0 or a real number [ ] 0,1 σ ∈ depending on whether the corresponding component of x λ is positive, negative, or zero.The generalized Newton method for finding a solution of the equation ( ) 0 f x λ = consists of the following iteration:

Theorem 3 :
generalized Newton iteration(3.6)converges to an accumulation point kx associated with k λ .First, we establish boundness of the sequence { } by the Newton iterates (3.6) and hence the existence of accumulation point at each generalized Newton iteration.Suppose that the interval matrix [ ] bounded.Consequently, there exits an accumulation points k

Theorem 4 :
Suppose that the interval matrix [ ] large k λ , then the generalized Newton iteration (3.6) linearly converges from any starting point 0 k x λ to a solution kx of the nonlinear penalized equation (3.1).

Supported
This work supported by the Science Foundation of Inner Mongolia in China (2011MS0114)