Abstract
By utilizing a dual complementarity condition, we propose an iterative method 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 a succession of linear programs. The algorithm was tested on 500 consecutively generated random solvable instances of the AVE with n = 10, 50, 100, 500 and 1,000. The algorithm solved 90.2 % of the test problems to an accuracy of 10−8.
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References
Chung S.-J.: NP-completeness of the linear complementarity problem. J. Optim. Theory Appl. 60, 393–399 (1989)
Cottle R.W., Dantzig G.: Complementary pivot theory of mathematical programming. Linear Algebra Appl. 1, 103–125 (1968)
Cottle R.W., Pang J.-S., Stone R.E.: The Linear Complementarity Problem. Academic Press, New York (1992)
Hu S.-L., Huang Z.-H.: A note on absolute equations. Optim. Lett. 4(3), 417–424 (2010)
ILOG, Incline Village, Nevada. ILOG CPLEX 9.0 User’s Manual, 2003. http://www.ilog.com/products/cplex/
Mangasarian O.L.: Linear complementarity problems solvable by a single linear program. Math. Program. 10, 263–270 (1976)
Mangasarian, O.L.: Absolute value equation solution via concave minimization. Optim. Lett. 1(1), 3–8 (2007). ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/06-02.pdf
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
Mangasarian, O.L.: A generalized Newton method for absolute value equations. Technical Report 08-01, Data Mining Institute, Computer Sciences Department, University of Wisconsin (2008). ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/08-01.pdf . Optim. Lett. 3(1), 101–108 (2009). Online version: http://www.springerlink.com/content/c076875254r7tn38/
Mangasarian, O.L.: Primal–dual bilinear programming solution of the absolute value equation. Technical report, Data Mining Institute, Computer Sciences Department, University of Wisconsin, Madison, Wisconsin (2011). ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/11-01.pdf. Optimization Lett. (to appear)
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
MATLAB. User’s Guide. The MathWorks, Inc., Natick, MA 01760 (1994–2006). http://www.mathworks.com
Prokopyev O.A.: On equivalent reformulations for absolute value equations. Comput. Optim. Appl. 44(3), 363–372 (2009)
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
Rohn J.: An algorithm for solving the absolute value equation. Electron. J. Linear Algebra 18, 589–599 (2009)
Rohn J.: On unique solvability of the absolute value equation. Optim. Lett. 3, 603–606 (2009)
Rohn J.: A residual existence theorem for linear equations. Optim. Lett. 4(2), 287–292 (2010)
Rohn, J.: An algorithm for computing all solutions of an absolute value equation. Optim. Lett. (2011, to appear)
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Mangasarian, O.L. Absolute value equation solution via dual complementarity. Optim Lett 7, 625–630 (2013). https://doi.org/10.1007/s11590-012-0469-5
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DOI: https://doi.org/10.1007/s11590-012-0469-5