Iterative methods for solving variational inequalities in Euclidean space

Abstract

In this paper, we investigate the convergence properties of an iterative method for solving variational inequalities in Euclidean space. We show that under certain assumptions the method can be applied to variational inequalities defined over the common fixed point set of a given infinite family of cutter operators. The main step of our method consists in the computation of the metric projection onto a certain superhalf-space, which is constructed using the input data defining the problem. Moreover, in the case where the common fixed point set is a finite intersection of the fixed point sets of cutters for which Opial's closedness principle holds, we show that the iterative method can be easily combined with either sequential, composition or convex combination type methods. We also discuss ways of applying our method to more general classes of operators such as quasi-nonexpansive and demi-contractive ones.Mathematics Subject Classification. 47H05, 47H09, 47H10, 47J20, 47J25, 65K15.

Outer approximation methods for solving variational inequalities in Hilbert space

“…We illustrate the iterative method (25) in this case in Figure 2. By imposing conditions on {λ k } ∞ k=0 , following Fukushima [28] and others, see [20], [17] and [29], we replace the fixed step size λ in the gradient projection method (2) by a null, non-summable sequence. This condition is quite common in optimization theory and, in particular, appears in the context of the HSD method (19); see, for example, the papers by Yamada and Ogura [46], Hirstoaga [32], Aoyama and Kohsaka [2], Cegielski and Zalas [18,19] and Cegielski and Al-Musallam [16].…”

“…In this direction, Gibali et al [29] have recently considered VIs with a subset C outerly approximated by an infinite family of cutters T k : R n → R n in the sense…”

“…In the definition of H k the constant operator T was replaced by a sequence of opera- [29,Condition 3.6] which is somewhat related to weak regularity and therefore to the demi-closedness principle has been imposed on the family of T k 's. In particular, for C defined as the solution set of the common fixed point problem for a finite family of weakly regular cutters U i : R n → R n , i ∈ I = {1, .…”

“…This suggests that similar sufficient conditions for the strong convergence of (5) should apply to (3) in the infinite dimensional setting. We emphasize here that convergence results for the iterative method (3), to the best of our knowledge, have so far been established in Euclidean space only, also by imposing global conditions on F different than Lipschitz continuity; compare with [28,Assumption (c)], [20,Condition 5], [17,Condition 3.3] and [29,Condition 3.4].…”

“…In the present paper we assume that F is Lipschitz continuous and strongly monotone, and that C is outerly approximated (compare with (4)) by an infinite sequence of cutters T k : H → H. The main contribution of our paper is to provide sufficient conditions for the strong convergence of method (3) in a general real Hilbert space H (Theorem 3.1). In particular, for C defined as the solution set of the common fixed point problem with respect to a finite family of operators U i : H → H, following [29], we allow the T k 's to be defined either by cyclic, simultaneous or composition algorithmic operators. Moreover, we permit not only cyclic but also maximum proximity algorithmic operators which are more general than the remotes-set and most-violated constraint case.…”