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 i...

In this paper, an improved harmony search with chaos (HSCH) is presented for solving NP-hard absolute value equation (AVE) Ax |x| = b, where A is an arbitrary square matrix whose singular values exceed one. The simulation results in solving some given AVE problems demonstrate that the HSCH algorithm is valid and outperforms the classical HS algorithm (HS) and HS algorithm with differential muta...

Presented is an algorithm which for each A, B ∈ Rn×n and b ∈ Rn in a finite number of steps either finds a solution of the equation Ax+B|x| = b, or states existence of a singular matrix S satisfying |S − A| ≤ |B| (and in most cases also constructs such an S).

We propose a hybrid algorithm for solving the NP-hard absolute value equation (AVE): Ax−|x| = b, where A is an n×n square matrix. The algorithm makes no assumptions on the AVE other than solvability and consists of solving iteratively a linear system of equations followed by a linear program. The algorithm was tested on 100 consecutively generated random solvable instances of the AVE with n =50...

In this paper, we reformulate the NP-hard problem of the absolute value equation (AVE) as a horizontal linear complementarity one and then solve it using a smoothing technique. This approach leads to a new class of methods that are valid for general absolute value equation. An asymptotic analysis proves the convergence of our schemes and provides some interesting error estimates. This kind of e...