Non-quadratic Proximal Regularization with Application to Variational Inequalities in Hilbert Spaces

We consider a generalized proximal point method 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 operator (using the -enlargement concept) and 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.