## 1 Introduction
On the other hand, the authors of [16; 18; 19] studied the generalized Fischer-Burmeister function, i.e., \(\phi_{p}:\Re^{n}\to\Re\) given by \(\phi_{p}(a,b)=\|(a,b)\|_{p}-(a+b)\)\((p\in(1,+\infty))\). Additionally, their research work has shown that this class of functions enjoys some favorable properties as other NCP-functions. The given numerical results for the test problems from mixed complementarity problem library (MCPLIB) have shown that the descent method has better performance when \(p\) decreases in \((1,+\infty)\). The main motivation of this paper is to use the advantages of [12; 15; 16; 18; 19] to solve the stochastic linear complementary problem denoted as (3). Firstly, we put forward a new robust reformulation of the complementarity Problem (3). Then, based on the new robust reformulation, we propose a feasible nonsmooth Levenberg-Marquardt-type method to solve (3). The numerical results in Section 4 showed that the given Method 1 is efficient for the related refinery production problem and the large-scale stochastic linear complementarity problems. We also make a comparison with Method 1 and the scaled trust region method (STRM) in [20]; preliminary numerical experiments showedthat the numerical results of Method 1 are as good as the numerical results of the STRM method. Additionally, it generates less iterations in contrast to the STRM method. Additionally, we also make a comparison with Method 1 and the method in [13] for solving the related refinery production problem. The preliminary numerical experiments also indicate that Method 1 is quite robust. In other words, Method 1 has the following advantages. * Faster reduction of the complementarity gap \(x^{T}(\bar{M}x+\bar{q})\). * Flexible NCP functions \(\phi_{p}\). * Randomly-chosen initial points and less calculation. Now is the time to give a new reformulation of (4); the new reformulation of (4) is:  
\[\phi_{n}(x)=\left( \begin{array}{c}\lambda\phi_{p}(x_{i},\bar{M}_{i}x+\bar{q }_{i}),i=1\cdots n\\ \vdots\\ (1-\lambda)\phi_{+}(x_{i},\bar{M}_{i}x+\bar{q}_{i}),i=1\cdots n\end{array}  \right), \tag{6}\]  
where \(\phi_{n}:\Re^{n}\rightarrow\Re^{2n}\), \(\lambda\in(0,1)\). Then, \(x\) is a solution of (4) \(\Longleftrightarrow\phi_{n}(x)=0\). Additionally, solving (4) is also equivalent to finding a solution of the unconstrained optimization problem:  
\[\min_{x\in\Re^{n}}\Psi(x), \tag{7}\]  
where:  
\[\Psi(x)=\frac{1}{2}\|\phi_{n}(x)\|^{2}. \tag{8}\]  
Then, (4) and (5) can be rewritten as:  
\[F(x,y)=0,y\geq 0, \tag{9}\]  
where:  
\[F(x,y)=\left( \begin{array}{c}\phi_{n}(x)\\ M(\omega_{1})x+q(\omega_{1})-y_{1}\\ M(\omega_{2})x+q(\omega_{2})-y_{2}\\ \vdots\\ M(\omega_{m})x+q(\omega_{m})-y_{m}\end{array} \right).\]  
Additionally, \(y=[y_{1}^{T},y_{2}^{T},...,y_{m}^{T}]^{T}\in\Re^{mn}\) is a slack variable with \(y_{i}\in\Re^{n},\ i=1,2,...,m\). Then, we know that \(F(x,y)=0\) has \((m+2)n\) equations with \((m+1)n\) variables. The remainder of this paper is organized as follows. In Section 2, we review some background definitions and some useful properties. In Section 3, we define a merit function \(\theta(z)=\frac{1}{2}\|F(z)\|^{2}\) and give a feasible nonsmooth Levenberg-Marquardt-type method. Some discussions about this method are also given. In Section 4, the numerical results indicate that the given method is efficient for solving stochastic linear complementarity problems, such as the related refinery production problem and the large-scale stochastic linear complementarity problems.