Regularization of Monotone Variational Inequalities with Mosco Approximations of the Constraint Sets

Abstract

In this paper we study the convergence and stability in reflexive, smooth and strictly convex Banach spaces of a regularization method for variational inequalities with data perturbations. We prove that, when applied to perturbed variational inequalities with monotone, demiclosed, convex valued operators satisfying certain conditions of asymptotic growth, the regularization method we consider produces sequences which converge weakly to the minimal-norm solution of the original variational inequality, provided that the perturbed constraint sets converge to the constraint set of the original inequality in the sense of a modified form of Mosco convergence of order 1. If the underlying Banach space has the Kadeč-Klee property, then the sequence generated by that regularization method is strongly convergent. Mathematics Subject Classifications (2000): Primary: 47J0G, 47A52; secondary: 47H14, 47J20.Key words: monotone operator, demiclosed operator, variational inequality, regularization method, Mosco convergence, fast Mosco-convergence relative to a sequence of positive real numbers.

Limit vector variational inequality problems via scalarization

“…Denote by x a point in int(D). From Lemma 2 in [2] (see also [1]), for any q ∈ P (C * ), there exist positive numbers r q = r q (x) and c q = c q (x) such that…”

A Proximal-Projection Method for Finding Zeros of Set-Valued Operators

“…These methods inspired the construction of a plethora of algorithms for finding zeros of various operators as well as for other purposes. Among them are the algorithms presented in [1], [3], [4], [6], [7], [8], [9], [10], [11], [12], [13], [14], [17], [20], [29], [35], [36], [69] which, in turn, inspired this research. The main formal differences between the proximal-projection method (1.7) and its already classical counterparts developed in the 50-ies and 60-ies consist of the use of the proximal projections instead of metric projections and of projecting not on the set C involved in Problem 1.1, but on some convex approximations C k of it.…”

“…They mostly are relaxed forms of Hausdorff metric convergence. It was shown in [8] that, in some circumstances, fast Mosco convergence of the sets C k to C (see [8,Definition 2.1]), a form of convergence significantly less demanding than Hausdorff convergence, is sufficient to make the proximal-projection method (1.7) applied to variational inequalities converge. As we show below, these convergence requirements in the case of the proximal-projection method (1.7) can be further weakened.…”