Using vector divisions in solving the linear complementarity problem

Abstract

The linear complementarity problem $LCP(M,q)$ is to find a vector $z$ in ${\mathrm{IR}}^n$ satisfying $z^T(Mz+q)=0$, $Mz+q⩾0$,$z⩾0$, where $M=\left(m_{ij}\right)\in {\mathrm{IR}}^{n\times n}$ and $q\in {\mathrm{IR}}^n$ are given. In this paper, we use the fact that solving $LCP(M,q)$ is equivalent to solving the nonlinear equation $F\left(x\right)=0$ where $F$ is a function from ${\mathrm{IR}}^n$ into itself defined by $F\left(x\right)=(M+I)x+(M-I)\left|x\right|+q$. We build a sequence of smooth functions $\tilde{F}(p,x)$ which is uniformly convergent to the function $F\left(x\right)$. We show that, an approximation of the solution of the $LCP(M,q)$ (when it exists) is obtained by solving $\tilde{F}(p,x)=0$ 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 $LCP(M,q)$. 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.