## 2 Preliminaries  
In this section, we give some related definitions and some properties; some of them can be found in [6; 14; 15; 19; 20; 21; 22; 23]; some of them are given for the first time. Let \(G:\Re^{m}\longrightarrow\Re^{n}\) be a locally-Lipschitzian function. The B-subdifferential of \(G\) at \(x\) is:  
\[\partial_{B}G(x)=\{V\in\Re^{m\times n}|\exists\{x_{k}\}\subseteq D_{G}:\{x_{ k}\}\to x,G^{\prime}(x_{k})\to V\},\]  
where \(D_{G}\) is the differentiable points set and \(G^{\prime}(x)\) is the Jacobian of \(G\) at a point \(x\in\Re^{n}\). The Clarke generalized Jacobian of \(G\) is defined as:  
\[\partial G(x)=conv\{V\in\Re^{m\times n}|\exists\{x_{k}\}\subseteq D_{G}:\{x_{k}\} \to x,G^{\prime}(x_{k})\to V\}.\]  
Furthermore,  
\[\partial_{C}G(x)^{T}=\partial G_{1}(x)\times\cdots\times\partial G_{m}(x)\]  
denotes the C-subdifferential of \(G\) at \(x\). If  
\[\lim_{V\in\partial G(x+Ht^{\prime}),H^{\prime}\to h,t\to 0^{+}}Vh^{\prime}\]  
exists for any \(h\in\Re^{m}\), we call \(G\) is semi-smooth at \(x\). Definition 1: ([6]) Matrix \(M\in\Re^{n\times n}\) is called a:  
1. \(P_{0}\)-matrix, if each of its principal minors is nonnegative. 2. \(P\)-matrix, if each of its principal minors is positive. Definition 2: ([23]) Let \(G:\Re^{n}\longrightarrow\Re^{n}\); the following statements are given:  
1. We call \(G\) monotone, if \((x-y)^{T}(G(x)-G(y))\geq 0\), for \(x,y\in\Re^{n}\). 2. \(G\) is a \(P_{0}\) function, if: \[\max_{1\leq i\leq n}\big{(}x_{i}-y_{i}\big{)}(G_{i}(x)-G_{i}(y))\geq 0\] for all \(x,y\in\Re^{n}\), \(x\
eq y\) and \(x_{i}\
eq y_{i}\). Proposition 1: ([6]) \(M\in\Re^{n\times n}\) is a \(P_{0}\)-matrix \(\Longleftrightarrow\forall x\
eq 0\), \(x_{i}(Mx)_{i}\geq 0,x_{i}\
eq 0\). Proposition 2: ([21]) Suppose \(G\) is a locally-Lipschitzian function and strongly semi-smooth at \(x\). Additionally, it is also directionally differentiable in a neighborhood of \(x\); we get:  
\[\lim_{h\to 0,H\in\partial G(x+h)}\frac{\|G(x+h)-G(x)-Hh\|}{\|h\|^{2}}<\infty.\]  
Definition 3: We call \(x^{*}\) an \(R\)-regular solution of the complementarity problem \(x\geq 0,G(x)\geq 0,x^{T}G(x)=0\), if \(G^{\prime}(x^{*})_{\alpha\alpha}\) is nonsingular and \(G^{\prime}(x^{*})_{\beta\beta}-G^{\prime}(x^{*})_{\beta\alpha}G^{\prime}(x^{*} )_{\alpha\delta}^{-1}G^{\prime}(x^{*})_{\alpha\delta}\) is a \(P\)-matrix, where \(\alpha=\{i|x^{*}_{i}>0,G_{i}(x^{*})=0\}\), \(\beta=\{i|x^{*}_{i}=0,G_{i}(x^{*})=0\}\), \(\gamma=\{i|x^{*}_{i}=0,G_{i}(x^{*})>0\}\). Proposition 3: ([15]) The generalized gradient of \(\phi_{P}\) at a point \((a,b)\) is defined as:  
\[\partial\phi(a,b)=(g_{a},g_{b})=\left\{ \begin{array}{c}(\frac{sgn(a)|a|^{p- 1}}{\|(a)\|_{p}^{p-1}}-1,\frac{sgn(b)|b|^{p-1}}{\|(a)\|_{p}^{p-1}}-1),(a,b) \
eq(0,0)\\ (\epsilon-1,\zeta-1),(a,b)=(0,0)\end{array} \right.,\]  
where \(\|(\epsilon,\zeta)\|_{P}\leq 1\). The generalized gradient of \(\phi_{+}\) at a point \((a,b)\) is defined as \(\partial\phi_{+}(a,b)=\{(b_{+}\partial a_{+},a_{+}\partial b_{+})\}\), where:  
\[\partial Z_{+}=\left\{ \begin{array}{c}1,Z>0\\ 0,Z<0\end{array} \right.,\]  
\(\partial Z_{+}=[0,1]\), if \(Z=0\). Definition 4: When \(x^{*}>0\), \(\bar{M}x^{*}+\bar{q}>0\), then (4) is called strictly feasible at \(x^{*}\). **Proposition 4**. : _([15]) All \(H\in\partial_{C}\phi(x)\) can be defined as:_  
\[\left( \begin{array}{c}\lambda H_{1}\\ (1-\lambda)H_{2}\end{array} \right),\]  
_where \(H_{1}\subseteq diag\{a_{i}(x)\}+diag\{b_{i}(x)\}\bar{M}\), \(H_{2}\subseteq diag\{\tilde{a}_{i}(x)\}+diag\{\tilde{b}_{i}(x)\}\bar{M}\), \((a_{i}(x),b_{i}(x))\in\partial\phi_{P}(x_{i},(\bar{M}x+\bar{q})_{i})\), \((\tilde{a}_{i}(x),\tilde{b}_{i}(x))\in\partial\phi_{+}(x_{i},(\bar{M}x+\bar{q} )_{i})\), \(\partial\phi_{P}(x_{i},(\bar{M}x+\bar{q})_{i})\) and \(\partial\phi_{+}(x_{i},(\bar{M}x+\bar{q})_{i})\) are given in Proposition 3._  
**Proposition 5**.