Optimization Letters

, Volume 1, Issue 1, pp 3–8

Absolute value equation solution via concave minimization

Original Paper

DOI: 10.1007/s11590-006-0005-6

Cite this article as:
Mangasarian, O.L. Optimization Letters (2007) 1: 3. doi:10.1007/s11590-006-0005-6


The 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.


Absolute value equation Concave minimization Successive linear programming 

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Computer Sciences DepartmentUniversity of WisconsinMadisonUSA
  2. 2.Department of MathematicsUniversity of California at San DiegoLa JollaUSA

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