## 3 The Feasible Nonsmooth Levenberg-Marquardt-Type Method and Its Convergence Analysis  
In this section, we define a merit function \(\theta(z)=\frac{1}{2}\|F(z)\|^{2}\) and give a feasible nonsmooth Levenberg-Marquardt-type method. At the same time, we also give some discussions aboutthis method. Let \(z=(x,y)\in\Re^{(m+1)n}\); define a merit function of (9) by:  
\[\theta(z)=\frac{1}{2}\|F(z)\|^{2}.\]  
If (3) has a solution, then solving (9) is equivalent to finding a global solution of the following constrained optimization problem:  
\[\min \theta(z)\] \[s.t. z\geq 0. \tag{10}\]  
If \(z\) satisfies:  
\[P_{Z}[z-\
abla\theta(z)]-z=0, \tag{11}\]  
where \(P_{Z}(\cdot)\) is an orthogonal projection operator onto \(Z=\{z\in\Re^{(m+1)n}|z\geq 0\}\), then \(z\) is a stationary point of (10). Obviously, (11) is equivalent to the following problem:  
\[\
abla\theta(z)\geq 0,z\geq 0,z^{T}\
abla\theta(z)=0. \tag{12}\]  
**Lemma 1**. : _([14]) Let \(P_{Z}(\cdot)\) be an orthogonal projection operator onto \(Z=\{z\in\Re^{(m+1)n}|z\geq 0\}\). The following statements hold:_  
1. \(\|P_{Z}(x)-P_{Z}(y)\|\leq\|x-y\|\) _for all_ \(x,y\in\Re^{(m+1)n}\)_._
2. _For any_ \(y\in Z\)_,_ \((P_{Z}(x)-x)^{T}(P_{Z}(x)-y)\leq 0\) _for all_ \(x\in\Re^{(m+1)n}\)_._  
**Proposition 9**. : _([15]) The merit function \(\theta\) has the following properties._  
1. \(\theta(z)\) _is continuously differentiable on_ \(\Re^{(m+1)n}\) _with_ \(\
abla\theta(z)=H^{T}F(z)\) _for any_ \(H\in\partial_{\mathbb{C}}F(z)\)_._
2. _Assume_ \(x\longmapsto\bar{M}x+\bar{q}\) _is monotone, if_ \(LCP(\bar{M},\bar{q})\) _has a strictly feasible solution, then for all_ \(c>0\)_, we know that the level set:_ \[L(c)=\{z\in\Re^{(m+1)n}|\theta(z)\leq c\}\] _is bounded._  
For some monotone stochastic linear complementarity problems, the stationary points of (10) may not be a solution. Such as let \(n=1\), \(m=2\), \(\Omega=\{\omega_{1},\omega_{2}\}=\{0,1\}\), \(M(\omega_{1})=M(\omega_{2})=1\), \(q(\omega_{1})=1\), \(q(\omega_{2})=-1\), and \(p_{i}=P\{\omega_{i}\in\Omega\}=0.5,i=1,2\), (see [12] ). By simple computation, we know that the above of problem is a monotone SLCP, obviously; all points \(x\geq 1\) are feasible, but this example has no solution. By:  
\[F(x,y_{1},y_{2})=\left( \begin{array}{c}\lambda(2x^{p})^{\frac{1}{p}}-2|x|\\ (1-\lambda)(x_{+})^{2}\\ x+1-y_{1}\\ x-1-y_{2}\end{array} \right),\]  
and \((0,1,0)\) is a stationary point of the constraint optimization problem:  
\[\min\frac{1}{2}\|F(x,y_{1},y_{2})\|^{2}\]  
\[s.t. x\geq 0,y_{1}\geq 0,y_{2}\geq 0.\]  
However, \(x=0\) is not a solution of this example. Therefore, in the following proposition, we give some conditions for (3). **Proposition 10**. : _For monotone Problem (3), let \(z^{*}=(x^{*},y^{*})\) be a stationary point of (10). If \(M(\omega_{i})x^{*}+q(\omega_{i})-y^{*}_{i}=0,i=1,2,...m\), then \(x^{*}\) is a solution of (3)._  
**Proof of Proposition 10**. : Assuming that \(z^{*}=(x^{*},y^{*})\) is a stationary point of (10), if \(M(\omega_{i})x^{*}+q(\omega_{i})-y^{*}_{i}=0,i=1,2,...m\), by (12), we know that \(x^{*}\) is the stationary point of the following problem:  
\[\min\{\Psi(x)|x\geq 0\}.\]  
Similar to the proof of Theorem 3 in [24], it can be shown that \(x^{*}\) is a solution of \(\Psi(x)=0\). Thus, \(x^{*}\) is a solution of (3). Now, we present the feasible nonsmooth Levenberg-Marquardt-type method for solving (3). **Method 1**. : _Choose \(z_{0}\in Z,\sigma\in(0,\frac{1}{2}),\varepsilon\geq 0,\beta,\gamma\in(0,1)\). Set \(k=0\)._  
_Step 1. If \(|\theta(z_{k})|\leq\varepsilon\), stop._  
_Step 2. Choose \(H_{k}\in\partial_{C}F(z_{k})\), \(\
u_{k}=\|F(z_{k})\|>0\), and find the solution \(d_{k}\) of the equations:_  
\[(H_{k}^{T}H_{k}+\
u_{k}I)d=-\
abla\theta(z_{k}). \tag{13}\]  
_Step 3. If_  
\[\|F(P_{Z}(z_{k}+d_{k}))\|\leq\gamma\|F(z_{k})\|,\]  
_then set \(z_{k+1}=P_{Z}(z_{k}+d_{k})\), \(k=k+1\), and go to Step 1; otherwise, go to Step 4._  
_Step 4.