Linear complementarity as absolute value equation solution

Abstract

We consider the linear complementarity problem (LCP): \(Mz+q\ge 0, z\ge 0, z^{\prime }(Mz+q)=0\) as an absolute value equation (AVE): \((M+I)z+q=|(M-I)z+q|\), where \(M\) is an \(n\times n\) square matrix and \(I\) is the identity matrix. We propose a concave minimization algorithm for solving (AVE) that consists of solving a few linear programs, typically two. The algorithm was tested on 500 consecutively generated random solvable instances of the LCP with \(n=10, 50, 100, 500\) and 1,000. The algorithm solved \(100\,\%\) of the test problems to an accuracy of \(10^{-8}\) by solving 2 or less linear programs per LCP problem.