Linear complementarity as absolute value equation solution

We consider the linear complementarity problem (LCP): Mz + q ≥ 0, z ≥ 0, z′(Mz + q) = 0 as an absolute value equation (AVE): (M + I)z + q = |(M − I)z + q|, where M is an n× 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.