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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)].

90C30 Nonlinear programming
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
65K05 Numerical mathematical programming methods
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