# 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)
A new inexact alternating directions method for monotone variational inequalities. (English) Zbl 1009.90108
Given real matrices $A$ of order $l×n$ and $B$ of order $l×m,$ let ${\Omega }=\left\{\left(x,y\right)|x\in X,y\in Y,Ax+By=b\right\}$ where $X$ and $Y$ are given nonempty closed convex subsets of ${ℝ}^{n}$ and ${ℝ}^{m},$ respectively, and $b$ is a given vector in ${ℝ}^{m}·$ Let $F\left(u\right)=\left(\begin{array}{c}f\left(x\right)\\ g\left(y\right)\end{array}\right),$ where $f:X\to {ℝ}^{n}$ and $g:Y\to {ℝ}^{m}$ are given monotone operators. This paper studies the variational inequality problem of determining a vector ${u}^{*}\in {\Omega }$ such that ${\left(u-{u}^{*}\right)}^{T}F\left({u}^{*}\right)\ge 0$ for all $u\in {\Omega }·$ An inexact alternating directions method for the above problem is presented that extends the method given in J. Eckstein, “Some saddle-function splitting methods for convex programming”, Optim. Methods Software 4, 75-83 (1994)].

##### MSC:
 90C30 Nonlinear programming 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions) 65K05 Mathematical programming (numerical methods)
##### Keywords:
variational inequality; alternating method; inexact method