Absolute value equation solution via concave minimization

The NP-hard absolute value equation (AVE) Ax − |x| = b where A ∈ R and b ∈ Rn is solved by a succession of linear programs. The linear programs arise from a reformulation of the AVE as the minimization of a piecewise-linear concave function on a polyhedral set and solving the latter by successive linearization. A simple MATLAB implementation of the successive linearization algorithm solved 100 consecutively generated 1000-dimensional random instances of the AVE with only five violated equations out of a total of 100, 000 equations.