Absolute value programming
 O. L. Mangasarian
 … show all 1 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
We investigate equations, inequalities and mathematical programs involving absolute values of variables such as the equation Ax+Bx = b, where A and B are arbitrary m× n real matrices. We show that this absolute value equation is NPhard to solve, and that solving it with B = I solves the general linear complementarity problem. We give sufficient optimality conditions and duality results for absolute value programs as well as theorems of the alternative for absolute value inequalities. We also propose concave minimization formulations for absolute value equations that are solved by a finite succession of linear programs. These algorithms terminate at a local minimum that solves the absolute value equation in almost all solvable random problems tried.
 Chung, S.J., Murty, K.G. Polynomially bounded ellipsoid algorithms for convex quadratic programming. In: Mangasarian, O.L., Meyer, R.R., Robinson, S.M. eds. (1981) Nonlinear Programming. Academic Press, New York, pp. 439485
 Cottle, R.W., Dantzig, G. (1968) Complementary pivot theory of mathematical programming. Linear Algebra and its Applications 1: pp. 103125 CrossRef
 Cottle, R.W., Pang, J.S., Stone, R.E. (1992) The Linear Complementarity Problem. Academic Press, New York
 CPLEX Optimization Inc., Incline Village, Nevada. Using the CPLEX(TM) Linear Optimizer and CPLEX(TM) Mixed Integer Optimizer (Version 2.0), 1992.
 O.L. Mangasarian, Nonlinear Programming, Reprint: SIAM Classic in Applied Mathematics 10, 1994, Philadelphia, McGraw–Hill, New York, 1969.
 Mangasarian, O.L. (1995) The linear complementarity problem as a separable bilinear program. J. Global Optim. 6: pp. 153161 CrossRef
 O.L. Mangasarian, “Solution of general linear complementarity problems via nondifferentiable concave minimization,” Acta Mathematica Vietnamica, vol. 22, no. 1, pp. 199–205, 1997, ftp://ftp.cs.wisc.edu/mathprog/techreports/9610.ps.
 Mangasarian, O.L., Meyer, R.R. (1979) Nonlinear perturbation of linear programs. SIAM Journal on Control and Optimization 17: pp. 745752 CrossRef
 MATLAB. User’s Guide. The MathWorks, Inc., Natick, MA 01760, 1994–2001. http://www.mathworks.com.
 Rockafellar, R.T. (1970) Convex Analysis. Princeton University Press, Princeton, New Jersey
 J. Rohn, “Systems of linear interval equations,” Linear Algebra and Its Applications, vol. 126, pp. 39–78, 1989. http://www.cs.cas.cz/~rohn/publist/47.doc.
 J. Rohn, “A theorem of the alternatives for the equation A x+Bx = b,” Linear and Multilinear Algebra, vol. 52, no. 6, pp. 421–426, 2004. http://www.cs.cas.cz/~rohn/publist/alternatives.pdf.
 Title
 Absolute value programming
 Journal

Computational Optimization and Applications
Volume 36, Issue 1 , pp 4353
 Cover Date
 20070101
 DOI
 10.1007/s1058900603955
 Print ISSN
 09266003
 Online ISSN
 15732894
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 Absolute value (AV) equations
 AV algorithm
 AV theorems of alternative
 AV duality
 Industry Sectors
 Authors

 O. L. Mangasarian ^{(1)}
 Author Affiliations

 1. Computer Sciences Department, University of Wisconsin, Madison, WI, 53706