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A new inexact alternating directions method for monotone variational inequalities. (English) Zbl 1009.90108
Given real matrices $$A$$ of order $$l\times n$$ and $$B$$ of order $$l\times m,$$ let $$\Omega =\{(x,y)|x\in X, y\in Y, Ax+By=b\}$$ where $$X$$ and $$Y$$ are given nonempty closed convex subsets of $$\mathbb{R}^n$$ and $$\mathbb{R}^m ,$$ respectively, and $$b$$ is a given vector in $$\mathbb{R}^m.$$ Let $$F(u)= \left ( \begin{matrix} f(x)\\ g(y) \end{matrix}\right),$$ where $$f: X\rightarrow \mathbb{R}^n$$ and $$g:Y\rightarrow \mathbb{R}^m$$ are given monotone operators. This paper studies the variational inequality problem of determining a vector $$u^*\in \Omega$$ such that $$(u-u^*)^TF(u^*)\geq 0$$ 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 and variational inequalities (finite dimensions) (aspects of mathematical programming) 65K05 Numerical mathematical programming methods
##### Keywords:
variational inequality; alternating method; inexact method
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