Using vector divisions in solving linear complementarity problem

The linear complementarity problem is to find vector $z$ in $\mathrm{IR}^{n}$ satisfying $z^{T}(Mz+q)=0$, $Mz+q\geqslant0,$ $z\geqslant0$, where $M$ as a matrix and $q$ as a vector, are given data; this problem becomes in present the subject of much important research because it arises in many areas and it includes important fields, we cite for example the linear and nonlinear programming, the convex quadratic programming and the variational inequalities problems, ... It is known that the linear complementarity problem is completely equivalent to solving nonlinear equation $F(x)=0$ with $F$ is a function from $\mathrm{IR}^{n}$ into itself defined by $F(x)=(M+I)x+(M-I)|x|+q$. In this paper we propose a globally convergent hybrid algorithm for solving this equation; this method is based on an algorithm given by Shi \cite{Y. Shi}, he uses vector divisions with the secant method; but for using this method we must have a function continuous with partial derivatives on an open set of $\mathrm{IR}^{n}$; so we built a sequence of functions $\tilde{F}(p,x)\in C^{\infty}$ which converges uniformly to the function $F(x)$; and we show that finding the zero of the function $F$ is completely equivalent to finding the zero of the sequence of the functions $\tilde{F}(p,x)$. We close our paper with some numerical simulation examples to illustrate our theoretical results.

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