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Using vector divisions in solving the linear complementarity problem. (English) Zbl 1238.65053
Summary: The linear complementarity problem $LCP\left(M,q\right)$ is to find a vector $z$ in ${ℝ}^{n}$ satisfying ${z}^{T}\left(Mz+q\right)=0$, $Mz+q\ge 0$, $z\ge 0$, where $M=\left({m}_{ij}\right)\in {ℝ}^{n×n}$ and $q\in {ℝ}^{n}$ are given. In this paper, we use the fact that solving $LCP\left(M,q\right)$ is equivalent to solving the nonlinear equation $F\left(x\right)=0$ where $F$ is a function from ${ℝ}^{n}$ into itself defined by $F\left(x\right)=\left(M+I\right)x+\left(M-I\right)x+\left(M-I\right)|x|+q$. We build a sequence of smooth functions $\stackrel{˜}{F}\left(p,x\right)$ which is uniformly convergent to the function $F\left(x\right)$. We show that, an approximation of the solution of the $LCP\left(M,q\right)$ (when it exists) is obtained by solving $\stackrel{˜}{F}\left(p,x\right)=0$ for a parameter $p$ large enough. Then we give a globally convergent hybrid algorithm which is based on vector divisions and the secant method for solving $LCP\left(M,q\right)$. We close our paper with some numerical simulations to illustrate our theoretical results, and to show that this method can solve efficiently large-scale linear complementarity problems.
##### MSC:
 65K05 Mathematical programming (numerical methods) 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions) 65H10 Systems of nonlinear equations (numerical methods)