Using vector divisions in solving the linear complementarity problem

Youssef Elfoutayeni (1) , Mohamed Khaladi (2)

1 LMDP - laboratoire de la Modélisation Informatique et Analyse Mathématiques des Systèmes Dynamiques
2 LMDP - laboratoire de la Modélisation Informatique et Analyse Mathématiques des Systèmes Dynamiques

The linear complementarity problem LCP(M,q) is to find a vector z in IRⁿ satisfying z^{T}(Mz+q)=0, Mz+q≥0, z≥0, where M=(m_{ij})∈IR^{n×n} and q∈IRⁿ are given. In this paper, we use the fact that solving LCP(M,q) is equivalent to solve the nonlinear equation F(x)=0 where F is a function from IRⁿ into itself defined by F(x)=(M+I)x+(M-I)|x|+q. We build a sequence of smooth functions F(p,x) uniformly convergent to the function F(x). We show that, an approximation of the solution of the LCP(M,q) (when it exists) is obtained by solving F(p,x)=0 for a parameter p large enough. Then we give a globally convergent hybrid algorithm based on vector divisions and the secant method for solving the LCP(M,q). We close our paper with some numerical simulation to illustrate our theoretical results, and to show that this method can solve efficiently large-scale linear complementarity problems.

Mots clés: secant method; Linear complementarity problem; vector division; global convergence; Newton's method; secant method.

Domaines: Recherche opérationnelle [math.OC]; Analyse numérique [math.NA]

Youssef Elfoutayeni, Mohamed Khaladi. Using vector divisions in solving the linear complementarity problem. 2010. ⟨hal-00482110⟩