# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Absolute value equations. (English) Zbl 1172.15302
Summary: We investigate existence and nonexistence of solutions for NP-hard equations involving absolute values of variables: $Ax -\vert x\vert = b$, where $A$ is an arbitrary $n\times n$ real matrix. By utilizing an equivalence relation to the linear complementarity problem (LCP) we give existence results for this class of absolute value equations (AVEs) as well as a method of solution for special cases. We also give nonexistence results for our AVE using theorems of the alternative and other arguments.

##### MSC:
 15A39 Linear inequalities of matrices
Full Text:
##### References:
 [1] Bennett, K. P.; Mangasarian, O. L.: Bilinear separation of two sets in n-space. Comput. optim. Appl. 2, 207-227 (1993) · Zbl 0795.90060 [2] Cottle, R. W.; Dantzig, G.: Complementary pivot theory of mathematical programming. Linear algebra appl. 1, 103-125 (1968) · Zbl 0155.28403 [3] Cottle, R. W.; Pang, J. -S.; Stone, R. E.: The linear complementarity problem. (1992) · Zbl 0757.90078 [4] Mangasarian, O. L.: Nonlinear programming. (1969) · Zbl 0194.20201 [5] Mangasarian, O. L.: Characterization of linear complementarity problems as linear programs. Math. program. Study 7, 74-87 (1978) · Zbl 0378.90053 [6] Mangasarian, O. L.: The linear complementarity problem as a separable bilinear program. J. global optim. 6, 153-161 (1995) · Zbl 0835.90102 [7] O.L. Mangasarian, Absolute value programming, Technical Report 05-04, Data Mining Institute, Computer Sciences Department, University of Wisconsin, Madison, WI, September 2005. Available from: <ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/05-04.ps>. Comput. Optim. Appl., in press.