Aviv Gibali

Algorithms for the Split Variational Inequality Problem

We propose a prototypical Split Inverse Problem (SIP) and a new variational problem, called the Split Variational Inequality Problem (SVIP), which is a SIP. It entails finding a solution of one inverse problem (e.g., a Variational Inequality Problem (VIP)), the image of which under a given bounded linear transformation is a solution of another inverse problem such as a VIP. We construct iterative algorithms that solve such problems, under reasonable conditions, in Hilbert space and then discuss special cases, some of which are new even in Euclidean space.

The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space

We present a subgradient extragradient method for solving variational inequalities in Hilbert space. In addition, we propose a modified version of our algorithm that finds a solution of a variational inequality which is also a fixed point of a given nonexpansive mapping. We establish weak convergence theorems for both algorithms.

Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space

We study two projection algorithms for solving the variational inequality problem in Hilbert space. One algorithm is a modified subgradient extragradient method in which an additional projection onto the intersection of two half-spaces is employed. Another algorithm is based on the shrinking projection method. We establish strong convergence theorems for both algorithms.

Extensions of Korpelevich's Extragradient Method for the Variational Inequality Problem in Euclidean Space

We present two extensions of Korpelevichs extragradient method for solving the variational inequality problem (VIP) in Euclidean space. In the first extension, we replace the second orthogonal projection onto the feasible set of the VIP in Korpelevichs extragradient method with a specific subgradient projection. The second extension allows projections onto the members of an infinite sequence of subsets which epi-converges to the feasible set of the VIP. We show that in both extensions the convergence of the method is preserved and present directions for further research.

An Algorithm for Solving the Variational Inequality Problem Over the Fixed Point Set of a Quasi-Nonexpansive Operator in Euclidean Space

This article is concerned with the variational inequality problem VIP(ℱ, Fix(T)): find such that for all z ∈ Fix(T), where T: ℝ n → ℝ n is quasi-nonexpansive, Fix(T) is its nonempty fixed point set, and ℱ: ℝ n → ℝ n is monotone. We propose, in particular, an algorithm which entails, at each step, projecting onto a suitably chosen half-space, and prove that the sequences it generates converge to the unique solution of the VIP. We also present an application of our result to a hierarchical optimization problem.

Outer approximation methods for solving variational inequalities in Hilbert space

Abstract In this paper, we study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operator F over a closed and convex set C. We assume that the set C can be outerly approximated by the fixed point sets of a sequence of certain quasi-nonexpansive operators called cutters. We propose an iterative method, the main idea of which is to project at each step onto a particular half-space constructed using the input data. Our approach is based on a method presented by Fukushima in 1986, which has recently been extended by several authors. In the present paper, we establish strong convergence in Hilbert space. We emphasize that to the best of our knowledge, Fukushima’s method has so far been considered only in the Euclidean setting with different conditions on F. We provide several examples for the case where C is the common fixed point set of a finite number of cutters with numerical illustrations of our theoretical results.

Note on the modified relaxation CQ algorithm for the split feasibility problem

In this paper we are concerned with the Split Feasibility Problem (SFP) in which there are given two Hilbert spaces

l 1 - l 2 regularization of split feasibility problems

H_1