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A combined relaxation method for a class of nonlinear variational inequalities. (English) Zbl 1013.49004

The author studies the following nonlinear variational inequality: \[ \left\{ \begin{matrix}\text{find} u^*\in U,\;\text{such that:}\\ \displaystyle \langle G(u^*),u-u^*\rangle+f(u)-f(u^*) \geq 0\;\;\;\forall u\in U, \end{matrix} \right. \] where \(U\subseteq V\subseteq\mathbb{R}^n\) are closed and convex, \(G\colon V\rightarrow\mathbb{R}^n\) is locally Lipschitz continuous and monotone and \(f\colon U\rightarrow\mathbb{R}^n\) is of the form \(f(u)=\max\limits_{i=1,\ldots,m}f_i(u)\), where \(f_i\colon\mathbb{R}^n\rightarrow\mathbb{R}\) are continuously differentiable convex functions. In the paper a class of combined relaxation methods for the above problem is described. Under some appropriate assumptions, a linear rate of convergence is proved.

MSC:

49J40 Variational inequalities
58E35 Variational inequalities (global problems) in infinite-dimensional spaces
49M25 Discrete approximations in optimal control

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