Convergence analysis of non-quadratic proximal methods for variational inequalities in Hilbert spaces

Authors:
Alexander Kaplan;Rainer Tichatschke
Affiliations:
Department of Mathematics, University of Trier, 54286 Trier, Germany;Department of Mathematics, University of Trier, 54286 Trier, Germany Corresponding author (E-mail: tichat@uni-trier.de)
Venue:
Journal of Global Optimization
Year:
2002

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Abstract

We consider a general approach for the convergence analysis of proximal-like methods for solving variational inequalities with maximal monotone operators in a Hilbert space. It proves to be that the conditions on the choice of a non-quadratic distance functional depend on the geometrical properties of the operator in the variational inequality, and –- in particular –- a standard assumption on the strict convexity of the kernel of the distance functional can be weakened if this operator possesses a certain `reserve of monotonicity'. A successive approximation of the `feasible set' is performed, and the arising auxiliary problems are solved approximately. Weak convergence of the proximal iterates to a solution of the original problem is proved.