3.5 Solutions obtained by the new algorithm
Linear complementarity problem (2.2) can be described as the form\(\ \min\left(V_{\tau}-\text{LV},V-g\right)=0\).
The discrete type of the above is
\(\min\left(BV^{{}^{\prime}}\left(\tau\right)-\ \text{BV}\left(\tau\right)-\text{Bw}\left(\tau\right)-(\tau),\ V(\tau)\ -\ g\right)=0,\)(3.6)
where\(\text{\ g}=\left[g_{1},g_{2},\cdots,g_{M-1}\right]^{T}\),\(\ g_{m}=\text{Kmax}\left(1-e^{x_{m}},0\right)\).
Vector\(\ a=\left[a_{1},a_{2},\cdots,a_{n}\right]^{T},b=\left[b_{1},b_{2},\cdots,b_{n}\right]^{T}\).
Let \(\min\left(a_{i},b_{i}\right)\) represents the component of vector\(\ \min\left(a,b\right)\).
Let\({\ V}^{n}=V\left(\tau_{n}\right)\), then the calculation of\(V^{n+1}\) can be obtained by solving the discrete problem\(\min\left(BV^{n+1}-\zeta^{n},V^{n+1}-g\right)=0\), where\(B=\left(I-\frac{k}{2}C+\frac{k^{2}}{12}C^{2}\right)\).
Let\({\ c}^{n}=c\left(\tau_{n}\right)\), then the vector\(\zeta^{n}\) can be obtained by the following formula
\(\zeta^{n}=\left(I+\frac{k}{2}C+\frac{k^{2}}{12}C^{2}\right)V^{n}+c^{n}\).
Let\(\mathcal{\ Q=}\left\{0,1\right\},M^{{}^{\prime}}=M-1\), considering the problem
\(\operatorname{}\left(B\left(\hat{\alpha}\right)V-\varpi\left(\hat{\alpha}\right)\right),\)(3.7)
where for\(\hat{\ \alpha}\epsilon\mathcal{Q}^{M^{{}^{\prime}}}\),\(B\left(\hat{\alpha}\right)\) is a monotone matrix of dimension\({\ M}^{{}^{\prime}}\), and \(\varpi\left(\hat{\alpha}\right)\) is a vector on\({\ M}^{{}^{\prime}}\).
For the case where\(\hat{\alpha}=\left({\hat{\alpha}}_{1},{\hat{\alpha}}_{2},\cdots,{\hat{\alpha}}_{M^{{}^{\prime}}}\right)\epsilon\mathcal{Q}^{M^{{}^{\prime}}}\)and \(B_{\text{ij}}\left(\hat{\alpha}\right)\) depend only on\({\hat{\ \alpha}}_{i}\), for each\(\hat{\ \alpha}\epsilon\mathcal{Q,}{\hat{\alpha}}^{\hat{a}}=\left(\hat{a},\hat{a},\cdots,\hat{a}\right)\in\mathcal{Q}^{M^{{}^{\prime}}},B^{\hat{\alpha}}=B\left({\hat{\alpha}}^{\hat{\alpha}}\right),\varpi^{\hat{\alpha}}=\varpi\left(\alpha^{\hat{\alpha}}\right)\), then (3.7) can be represented equivalently as following
\(\operatorname{}\left(B^{\hat{\alpha}}V-\varpi^{\hat{\alpha}}\right)\)(3.8)
Therefore, Eq. (3.6) can be written as Eq. (3.8) in the case\({\ B}^{0}=B,B^{1}=I,\varpi^{0}=\zeta^{n},\varpi^{1}=g\) .
Then this algorithm is applied to find the solution of (3.8).
(1) Initialize\({\hat{\alpha}}_{0}\in\mathcal{Q}^{M^{{}^{\prime}}}=\left\{0,1\right\}^{M^{{}^{\prime}}}\)
(2) For \(k\left(\geq 0\right)\) iteration
(a) Find\(\ V^{\left(k\right)}\in R^{M^{{}^{\prime}}}\), so that\(\ B\left({\hat{\alpha}}^{k}\right)V^{\left(k\right)}=\varpi\left({\hat{\alpha}}^{k}\right)\). If \(k\geq 1\)and\(\ V^{\left(k\right)}=V^{\left(k-1\right)}\), the iteration stops, otherwise do (b)
(b) For each\(\ i=1,2,\cdots,M^{{}^{\prime}}\), let\({\hat{\alpha}}_{i}^{k+1}=\left\{\par \begin{matrix}\&0,\text{if}\ \left(B^{0}V^{\left(k\right)}-\varpi^{0}\right)\leq\left(B^{1}V^{\left(k\right)}-\varpi^{1}\right)\\ \&1,\text{otherwise}\\ \end{matrix}\right.\ \)
(c) Set\(\ k=k+1\ \)and go back to (a)
This algorithm generates a series of approximate values \(V^{k}\) for\(V^{n+1}\) and if coefficient \(B\) is M-matrix, it has finite terminal. In all our numerical experiments, we have not encountered any violation of this nature.