Optimization Letters

, Volume 3, Issue 1, pp 101–108 | Cite as

A generalized Newton method for absolute value equations

Original Paper

Abstract

A direct generalized Newton method is proposed for solving the NP-hard absolute value equation (AVE) Ax − |x| = b when the singular values of A exceed 1. A simple MATLAB implementation of the method solved 100 randomly generated 1,000-dimensional AVEs to an accuracy of 10−6 in less than 10 s each. Similarly, AVEs corresponding to 100 randomly generated linear complementarity problems with 1,000 × 1,000 nonsymmetric positive definite matrices were also solved to the same accuracy in less than 29 s each.

Keywords

Absolute value equation Generalized Newton Linear complementarity problems 

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References

  1. 1.
    Bertsekas D.P.: Nonlinear Programming. Athena Scientific, Belmont (1995)MATHGoogle Scholar
  2. 2.
    Chung S.J.: NP-completeness of the linear complementarity problem. J. Optim. Theory Appl. 60, 393–399 (1989)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cottle R.W., Dantzig G.: Complementary pivot theory of mathematical programming. Linear Algebra Appl. 1, 103–125 (1968)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cottle R.W., Pang J.S., Stone R.E.: The Linear Complementarity Problem. Academic Press, New York (1992)MATHGoogle Scholar
  5. 5.
    Mangasarian, O.L.: Absolute value equation solution via concaveminimization. Optim. Lett. 1(1), 3–8 (2007). ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/06-02.pdf
  6. 6.
    Mangasarian, O.L.: Absolute value programming. Comput. Optim. Appl. 36(1), 43–53 (2007). ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/05-04.ps Google Scholar
  7. 7.
    Mangasarian, O.L., Meyer, R.R.: Absolute value equations. Linear Algebra Appl. 419, 359–367 (2006). ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/05-06.pdf Google Scholar
  8. 8.
    Ortega J.M., Rheinboldt W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)MATHGoogle Scholar
  9. 9.
    Polyak B.T.: Introduction to Optimization. Optimization Software Inc, Publications Division, New York (1987)Google Scholar
  10. 10.
    Rockafellar R.T.: New applications of duality in convex programming. In Proceedings Fourth Conference on Probability, Brasov (1971)Google Scholar
  11. 11.
    Rohn, J.: A theorem of the alternatives for the equation A x + B|x| = b. Linear Multilinear Algebra, 52(6), 421–426 (2004). http://www.cs.cas.cz/~rohn/publist/alternatives.pdf

Copyright information

© Springer-Verlag 2008

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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