Using vector divisions in solving linear complementarity problem

• Main Authors: Elfoutayeni, Youssef, Khaladi, Mohamed
• Format: Journal Article
• Language: English
• Published: 09-05-2010
• Subjects: Computer Science - Numerical Analysis; Mathematics - Numerical Analysis
• DOI: 10.48550/arxiv.1005.1417

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.