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Absolute value equation solution via concave minimization. (English) Zbl 1149.90098
Summary: The NP-hard absolute value equation (AVE) Ax-|x|=b where AR n×n and bR 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.
MSC:
90C05Linear programming
Software:
CPLEX; Matlab
References:
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[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.ps
[8]MATLAB: User’s guide. The MathWorks, Inc., Natick, MA 01760 (1994–2001) http://www.mathworks.com
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[10]Rockafellar R.T. (1970) Convex Analysis. Princeton University Press, Princeton
[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.pdf · Zbl 1070.15002 · doi:10.1080/0308108042000220686