An Singular Values Based Newton Method for Linear Complementarity Problems

The existence condition of the solution of special nonlinear penalized equation of the linear complementarity problems is obtained by the relationship between penalized equations and an abso-lute value equation. Newton method is used to solve penalized equation, and then the solution of the linear complementarity problems is obtained. We show that the proposed method is globally and superlinearly convergent when the matrix of complementarity problems of its singular values exceeds 0; numerical results show that our proposed method is very effective and efficient.


Introduction
Given a matrix n n A R × ∈ and a vector n b R ∈ , the problem of finding vectors is called the linear complementarity problem (LCP). We call the problem is the LCP (A, b). It is well known that several problems in optimization and engineering can be expressed as LCPs. Cottle, Pang, and Stone [1] [2] provide a thorough discussion of the problem and its applications, as well as providing solution techniques.
There are a large number of general purpose methods for solving linear complementarity problems. We can divide these methods into essentially two categories: direct methods, such as pivoting techniques [1] [2], and iterative methods, such as Newton iteration [2] [3] and interior point algorithms [4].

Generalized Newton Method
In this section, we will propose that a new generalized Newton method based on the nonlinear penalized Equation (1.2) for solving the linear complementarity problem. Proposition 1 [15].

Algorithm 1
Step 1: Choose an arbitrary initial point 0 0 n x R λ ∈ , 0 ε > and given 0 Step 3: If Step4: If is solution of LCP. Otherwise let We know subsequence is bounded. Hence, exists convergence subsequence and assume that convergence point is x  , and satisfy Since the singular values of A exceed 0, then A is regular, and A I λ + is regular, we know that ( ) Proof. Similar to the proof of Theorem 5 in [15]. □

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 , LCP A b of as follows (This example appears in the Geiger and Kanzow [16], Jiang and Qi [17], YONG Long-quan, DENG Fang-an, CHEN Tao [18] and Han [15]): 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.  Table 2. This 0 x is initial point, k is number of inner iterations, the outer iteration number is m, x * is iteration results.
Example 3: The matrix A of linear complementarity problem ( ) , LCP A b of as follows (This example appears in the Geiger and Kanzow [16], Jiang and Qi [17], YONG Long-quan, DENG Fang-an, CHEN Tao [18] and Han [15]): The computational results are shown in Table 3. This 0 x is initial point, k is number of inner iterations, the outer iteration number is m, x * is iteration results.