Absolute value programming

We investigate equations, inequalities and mathematical programs involving absolute values of variables such as the equation Ax + B|x| = b, where A and B are arbitrary m × n real matrices. We show that this absolute value equation is NP-hard to solve, and that solving it solves the general linear complementarity problem. However, we propose a concave minimization problem for its solution that is solvable by a simple successive linearization algorithm. This algorithm terminates at a local minimum which solves the absolute value equation in numerical tests tried when m ≥ 2n or n ≥ 2m. We also give sufficient optimality conditions and duality results for absolute value programs as well as theorems of the alternative for absolute value inequalities.