Abstract
This paper investigate the NP-hard absolute value equation (AVE) Ax - |x| = b, where A is an arbitrary square matrix whose singular values exceed one. The significance of the absolute value equation arises from the fact that linear programs, quadratic programs, bimatrix games and other problems can all be reduced to the linear complementarity problem that in turn is equivalent to the absolute value equation. This paper present a new smoothing function to the AVE. Based on this function, a smoothing Newton method is proposed for solving the AVE under the less stringent condition that the singular values of A exceed 1. The global convergence of the method is established under appropriate conditions. Preliminary numerical results indicate that this method is promising.
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