## 1 Introduction  
We consider the nonlinear nonsmooth complementarity problem, which is to find a vector in \(R^{n}\) satisfying the conditions  
\[x\geq 0,F(x)\geq 0,x^{T}F(x)=0\]  
where \(F:R^{n}\to R^{n}\) is a locally Lipschitz continuous function. If \(F\) is continuously differentiable, then Problem (1) is called the nonlinear complementarity problems NCP(F). As we all know, Equation (1) is a very useful general mathematics model, which is closely related to the mathematical programming, variational inequalities, fixed point problems and mixed strategy problems (such as [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13]). The methods for solving NCP(F) are classified into three categories: nonsmoothNewton methods, Jacobian smoothing methods and smoothing methods (see [14; 15; 16; 17; 18; 19]). Conjugate gradient methods are widely and increasingly used for solving unconstrained optimization problem, especially in large-scale cases. There are few scholar has investigated the problem how to use the conjugate gradient method to solve NCP(F) (such as [10,20]). Moreover, in these papers, \(F\) is required to be a continuous differentiable \(P_{0}+R_{0}\) function. In this paper, we present a new smoothing conjugate gradient method for solving Equation (1), where \(F\) is only required to be a locally Lipschitz continuous function. In this paper, we also define the generalized gradient of \(F\) at \(x\) is  
\[\partial F(x)=conv\{\lim_{x_{k}\to x,x_{k}\in D_{F}}\bigtriangledown F(x_{k})\}\]  
where \({}^{\prime\prime}conv^{\prime\prime}\) denotes the convex hull of a set, \(D_{F}\) denotes the set of points at which \(F\) is differentiable (see [21]). In the following, we introduce the definition of the smoothing function. **Definition 1** (see [22]) Let \(F:R^{n}\longrightarrow R^{n}\) be a locally Lipschitz continuous function. We call \(\widetilde{F}:R^{n}\times R_{+}\longrightarrow R^{n}\) is a smoothing function of \(F\), if \(\widetilde{F}(x,\mu)\) is continuously differentiable in \(R^{n}\) for any fixed \(\mu>0\), and  
\[\lim_{\mu\longrightarrow 0}\widetilde{F}(x,\mu)=F(x)\]  
for any fixed \(x\in R^{n}\). If  
\[\lim_{x_{k}\longrightarrow x,\mu\downarrow 0}\bigtriangledown\widetilde{F}(x_{k},\mu)\in\partial F(x)\]  
for any \(x_{k}\in R^{n}\), we say \(F\) satisfies gradient consistency property. In the following sections of our paper, we also use the Fischer-Burmeister function (see [23]) and the smoothing Fischer-Burmeister function. (1) The Fischer-Burmeister function  
\[\varphi(a,b)=\sqrt{a^{2}+b^{2}}-a-b,(a,b)^{T}\in R^{2}\]  
where \(\varphi:R^{2}\longrightarrow R\). From the definition of \(\varphi\), we know that it is twice continuously differentiable besides \((0,0)^{T}\). Moreover, it is a complementarity function, which satisfies  
\[\varphi(a,b)=0\Longleftrightarrow a\geq 0,b\geq 0,ab=0\]  
Denote  
\[H(x)=\left( \begin{array}{c}\varphi(x_{1},F_{1}(x))\\ \vdots\\ \varphi(x_{n},F_{n}(x))\end{array} \right)\]  
It is obvious that \(H(x)\) is zero at a point \(x\) if and only if \(x\) is a solution of Equation (1). Then Equation (1) can be transformed into the following unconstrained optimization problem  
\[\min\Psi(x)=\frac{1}{2}\|H(x)\|^{2}\]  
We know that the optimal value of \(\Psi\) is zero, and \(\Psi\) is called the value function of Equation (1). (2) The smoothing Fischer-Burmeister function  
\[\varphi_{\mu}(a,b)=\sqrt{a^{2}+b^{2}+\mu}-a-b\]where \(\varphi:R^{3}\longrightarrow R\) and \(\mu>0\). Let  
\[H_{\mu}(x)=\left( \begin{array}{c}\varphi_{\mu}(x_{1},\tilde{F}_{1}(x,\mu))\\ \vdots\\ \varphi_{\mu}(x_{n},\tilde{F}_{n}(x,\mu))\end{array} \right)\]  
\[\Psi_{\mu}(x)=\frac{1}{2}\|H_{\mu}(x)\|^{2}\]  
where \(\tilde{F}_{i}(x,\mu)\) is the smoothing function of \(F_{i}(x),i=1,\cdots,n\). The rest of this work is organized as follows. In Section 2, we describe the new smoothing conjugate gradient method for the solution of Problem (1). It is shown that this method has global convergence properties under fairly mild assumptions.