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Zbl 0979.49006
He, Bingsheng
Inexact implicit methods for monotone general variational inequalities. (English)
[J] Math. Program. 86, No.1 (A), 199-217 (1999). ISSN 0025-5610; ISSN 1436-4646/e

Summary: Solving a variational inequality problem is equivalent to finding a solution of a system of non-smooth equations. Recently, we proposed an implicit method, which solves monotone variational inequality problems via solving a series of systems of nonlinear smooth (whenever the operator is smooth) equations. It can exploit the facilities of the classical Newton-like methods for smooth equations. In this paper, we extend the method to solve a class of general variational inequality problems $$Q(u^*)\in \Omega,\qquad (v- Q(u^*))^T F(u^*)\ge 0,\qquad \forall v\in \Omega.$$ Moreover, we improve the implicit method to allow inexact solutions of the systems of nonlinear equations at each iteration. The method is shown to preserve the same convergence properties as the original implicit method.

MSC 2000:: *49J40 Variational methods including variational inequalities
90C30 Nonlinear programming
90C33 Complementarity problems
47J20 Inequalities involving nonlinear operators

Keywords: variational inequalities; non-smooth equations; implicit method; systems of nonlinear equations

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Zentralblatt MATH Berlin [Germany]