Using vector divisions in solving the linear complementarity problem

  • Authors:

  • Youssef Elfoutayeni;Mohamed Khaladi

  • Affiliations:

  • Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, Marrakech, Morocco and UMI UMMISCO, IRD - UPMC, France and Computer Sciences Department, School of Engineering and I ...;Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, Marrakech, Morocco and UMI UMMISCO, IRD - UPMC, France

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2012

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Abstract

The linear complementarity problem LCP(M,q) is to find a vector z in IR^n satisfying z^T(Mz+q)=0, Mz+q=0,z=0, where M=(m"i"j)@?IR^n^x^n and q@?IR^n are given. In this paper, we use the fact that solving LCP(M,q) is equivalent to solving the nonlinear equation F(x)=0 where F is a function from IR^n into itself defined by F(x)=(M+I)x+(M-I)|x|+q. We build a sequence of smooth functions F@?(p,x) which is 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 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.