A New Interior Point Method for Linear Complementarity Problem

Youssef Elfoutayeni (1) , Mohamed Khaladi (1)

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

For a given n-vector q and a real square matrix M∈IR^{n×n}, the linear complementarity problem, denoted LCP(M,q), is that of finding nonnegative vector z∈IRⁿ such that z^{T}(Mz+q)=0 and Mz+q≥0. In this paper we suppose that the matrix M must be a symmetric and positive definite and the set S={z∈ IRⁿ / z>0 and Mz+q>0}; named interior points set of the LCP(M,q) must be nonempty. The aim of this paper is to show that the LCP(M,q) is completely equivalent to a convex quadratic programming problem (CQPP) under linear constraints. To solve the second problem, we propose an iterative method of interior points which converge in polynomial time to the exact solution; this convergence requires at most o(n^{0,5}L) iterations, where n is the number of the variables and L is the length of a binary coding of the input; furthermore, the algorithm does not exceed o(n^{3,5}L) arithmetic operations until its convergence and in the end, we close our paper with some numerical examples which illustrate our theoretical results.

Mots clés: Interior Point Algorithm; Linear Complementarity Problem; Convex Quadratic Programming with Equilibrium Constraints; Matrix symmetric and positive definite; Interior Point Algorithm.

Domaines: Recherche opérationnelle [math.OC]

Youssef Elfoutayeni, Mohamed Khaladi. A New Interior Point Method for Linear Complementarity Problem. 2010. ⟨hal-00486207⟩