Computational Optimization and Applications

, Volume 36, Issue 1, pp 43-53

First online:

Absolute value programming

  • O. L. MangasarianAffiliated withComputer Sciences Department, University of Wisconsin Email author 

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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 with B = I solves the general linear complementarity problem. We give sufficient optimality conditions and duality results for absolute value programs as well as theorems of the alternative for absolute value inequalities. We also propose concave minimization formulations for absolute value equations that are solved by a finite succession of linear programs. These algorithms terminate at a local minimum that solves the absolute value equation in almost all solvable random problems tried.


Absolute value (AV) equations AV algorithm AV theorems of alternative AV duality