Absolute value equation solution via concave minimization

AbstractThe NP-hard absolute value equation (AVE) Ax − |x| = b where $$A\in R^{n\times n}$$ and $$b\in R^n$$ 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 1,000-dimensional random instances of the AVE with only five violated equations out of a total of 100,000 equations.