## 1 Introduction  
Suppose \((\Omega,F,P)\) is a probability space with \(\Omega\subseteq\Re^{n}\) and \(P\) is a known probability distribution. A stochastic linear complementarity problem takes the form:  
\[x\geq 0,M(\omega)x+q(\omega)\geq 0,x^{T}[M(\omega)x+q(\omega)]=0\quad a.e. \quad\omega\in\Omega, \tag{1}\]  
where \(\Omega\subseteq\Re^{n}\) is the underlying sample space, for given probability distribution \(P\) and \(\forall\ \omega\in\Omega\), \(M(\omega)\in\Re^{n\times m}\) and \(q(\omega)\in\Re^{n}\). (1) is denoted as SLCP \((M(\cdot),q(\cdot))\) or SLCP; see [1; 2; 3; 4]. If \(\Omega\) has only one element, (1) is the standard linear complementarity problem (LCP), which has been studied in [5; 6]. Generally, there is no \(x\) satisfying (1) for all \(\omega\in\Omega\). In order to obtain a reasonable solution of Problem (1), several types of models have been proposed (one can see [1; 2; 3; 4; 7; 8; 9; 10; 11; 12; 13; 14]). One of them is the expected value (EV) method (see [1]). The EV model is to find a vector \(x\in\Re^{n}\), such that:  
\[0\leq x\perp\bar{M}x+\bar{q}\geq 0, \tag{2}\]  
where \(\bar{M}=E[M(\omega)]\), \(\bar{q}=E[q(\omega)]\), and \(E[. ]\) is the mathematical expectation. Another is the expected residual minimization (ERM) method (see [2]). The ERM model is to find a vector \(x\in\Re^{n}_{+}\) that minimizes the expected residual function:  
\[\min_{x\geq 0}\sum_{i=1}^{n}E[\varphi(x_{i},M_{i}(\omega)x+q_{i}(\omega))]^{2},\]where \(M_{i}(\omega)(i=1,\cdots,n)\) is the \(i\)-th line of random matrix \(M(\omega)\) and \(\varphi\) satisfies:  
\[\varphi(a,b)=0\Longleftrightarrow a\geq 0,b\geq 0,ab=0.\]  
In addition to [1; 2], [PERSON] and [PERSON] first considered a quasi-Monte Carlo method in [8; 9]; they proved that the ERM method is convergent under mild conditions and studied the properties of the ERM problems. In [10], [PERSON], [PERSON] and [PERSON] considered SLCP including the expectation of matrix is positive semi-definite. Under the condition of a new error bound, they use the ERM formulation to solve the SLCP. In [11], they also studied the ERM formulation, where the involved function is a stochastic \(R_{0}\) function. In [12], [PERSON] and [PERSON] put forward a new model of the stochastic linear complementarity problem with only finitely many elements. A feasible semi-smooth Newton method was also given. In [14], [PERSON] also considered a feasible semi-smooth Gauss-Newton method for solving this new stochastic linear complementarity problem. The stochastic linear complementarity problem with only finitely many elements is to find a vector \(x\in\Re^{n}\), such that:  
\[x\geq 0,M(\omega_{i})x+q(\omega_{i})\geq 0,x^{T}[M(\omega_{i})x+q(\omega_{i})] =0\quad i=1,\cdots,m,\quad m>1, \tag{3}\]  
where \(\Omega=\{\omega_{1},\omega_{2},\cdots,\omega_{m}\}\). Denote:  
\[\bar{M}=\sum_{i=1}^{m}p_{i}M(\omega_{i}),\bar{q}=\sum_{i=1}^{m}p_{i}q(\omega_ {i}),\]  
where \(p_{i}=P(\omega_{i}\in\Omega)>0,i=1,\cdots,m\). Then, (3) is equivalent to:  
\[x\geq 0,\bar{M}x+\bar{q}\geq 0,x^{T}(\bar{M}x+\bar{q})=0, \tag{4}\]  
\[M(\omega_{i})x+q(\omega_{i})\geq 0,\quad i=1,\cdots,m. \tag{5}\]  
where (4) is called the linear complementarity problem. As we all know, in [15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29], many methods were given for solving the nonlinear complementarity problem (NCP) and linear complementarity problem (LCP), such as [PERSON] and [PERSON], who presented a nonsmooth least squares reformulation of (4) in [15]. They defined \(\Phi:\Re^{n}\to\Re^{2m}\) as:  
\[\Phi(x)=\left( \begin{array}{c}\lambda\phi_{FB}(x_{i},\bar{M}_{i}x+\bar{q}_{i }),i=1\cdots n\\ \vdots\\ (1-\lambda)\phi_{+}(x_{i},\bar{M}_{i}x+\bar{q}_{i}),i=1\cdots n\end{array}  \right),\]  
where \(\lambda\in(0,1)\), \(\phi_{FB}(a,b)=\|(a,b)\|_{2}-(a+b)\), \(\phi_{+}(a,b)=a_{+}b_{+},z_{+}=max\{0,z\}\) for \(z\in\Re\). This least squares formulation can gave a faster reduction of the complementarity gap \(x^{T}(\bar{M}x+\bar{q})\).