Iterative methods for a class of linear complementarity problems

Abstract

This paper is concerned with two cases of iterative methods for solving linear complementarity problems (LCPs). First one is concerned with a specific LCP; find $x=(x_I,\;x_J)$ in $R_+^n$ and y in $R_+^m$ such that $c_I+D_Ix+y\ge0,\;\;c_J+D_Jx\ge0,\;\;b-x_I0$, \;\;x_I^T\;(c_I+D_Ix+y)=x_J^T(c_J+D_Jx)=0,\;\; y^T(b-x_I)=0$, and $b\ge0$. This type of problems can be found in a multiproduct market equilibrium model with restricted institutional regulations. It is shown that if $D_{JJ}$ part of D is strictly copositive, existence of a solution to this problem is guaranteed, and that the solution is unique if D is a P-matrix. An iterative algorithm for the problem, which is not new, but an extension of Mangasarian``s and Ahn;s, is developed. Its convergence properties are discussed. For symmetric cases, if $D_{JJ}$ part is strictly copositive, or copositive plus with some qualifications, convergence is attained. For nonsymmetric cases, where D is a Z-matrix, or an H-matrix with positive diagonals, convergence is also guaranteed with proper choice of relaxation parameter. Secondly, an iterative algorithm which is useful for the general LCP (M, q) where M is nonsymmetric positive definite matrix is suggested. It sconvergence properties are investigated.