Optimization Letters

, Volume 1, Issue 1, pp 3–8

Absolute value equation solution via concave minimization

Original Paper

Abstract

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.

Keywords

Absolute value equation Concave minimization Successive linear programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chung S.-J. (1989) NP-completeness of the linear complementarity problem. J. Optim. Theory Appl. 60, 393–399MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Cottle R.W., Dantzig G. (1968) Complementary pivot theory of mathematical programming. Linear Algebra Appl. 1, 103–125MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Cottle R.W., Pang J.-S., Stone R.E. (1992) The linear complementarity problem. Academic, New YorkMATHGoogle Scholar
  4. 4.
    ILOG Incline Village, Nevada. ILOG CPLEX 9.0 User’s Manual (2003) http://www.ilog.com/products/cplex/Google Scholar
  5. 5.
    Mangasarian O.L. (1997) Solution of general linear complementarity problems via nondifferentiable concave minimization. Acta Math. Vietnam. 22(1): 199–205 ftp://ftp.cs.wisc.edu/math-prog/tech-reports/96-10.psMATHMathSciNetGoogle Scholar
  6. 6.
    Mangasarian, O.L Absolute value programming. Technical Report 05-04, Data Mining Institute, Computer Sciences Department, University of Wisconsin, Madison, Wisconsin, September (2005) ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/05-04.ps. Comput. Optim. Appli. (to appear)Google Scholar
  7. 7.
    Mangasarian, O.L., Meyer, R.R. Absolute value equations. Technical Report 05–06, Data Mining Institute, Computer Sciences Department, University of Wisconsin, Madison, Wisconsin, December 2005. Linear Algebra and Its Applications (to appear) ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/05-06.psGoogle Scholar
  8. 8.
    MATLAB: User’s guide. The MathWorks, Inc., Natick, MA 01760 (1994–2001) http://www.mathworks.comGoogle Scholar
  9. 9.
    Polyak B.T. (1987) Introduction to optimization. Optimization Software, Inc., Publications Division, New YorkMATHGoogle Scholar
  10. 10.
    Rockafellar R.T. (1970) Convex Analysis. Princeton University Press, PrincetonMATHGoogle Scholar
  11. 11.
    Rohn J. (2004) A theorem of the alternatives for the equation A x + B|x| = b. Linear Multilinear Algebra 52(6): 421–426 http://www.cs.cas.cz/rohn/publist/alternatives.pdfMATHMathSciNetCrossRefGoogle Scholar

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

Personalised recommendations