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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 $A\in {R}^{n×n}$ 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.
##### MSC:
 90C05 Linear programming
CPLEX; Matlab
##### References:
 [1] Chung S.-J. (1989) NP-completeness of the linear complementarity problem. J. Optim. Theory Appl. 60, 393–399 · Zbl 0632.90072 · doi:10.1007/BF00940344 [2] Cottle R.W., Dantzig G. (1968) Complementary pivot theory of mathematical programming. Linear Algebra Appl. 1, 103–125 · Zbl 0155.28403 · doi:10.1016/0024-3795(68)90052-9 [3] Cottle R.W., Pang J.-S., Stone R.E. (1992) The linear complementarity problem. Academic, New York [4] ILOG Incline Village, Nevada. ILOG CPLEX 9.0 User’s Manual (2003) http://www.ilog.com/products/cplex/ [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.ps [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) [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 [9] Polyak B.T. (1987) Introduction to optimization. Optimization Software, Inc., Publications Division, New York [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