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Smooth Newton method to absolute value equation based on lower uniform smoothing approximation function

Abstract We 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 equations 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. In this paper, after a lower uniform smooth approximation function of absolute value function is established, absolute value equation is transformed into smooth optimization problem, and solved by smoothing Newton method. The convergence of method is shown, and the preliminary numerical experiments are given.