Algorithms for the Split Variational Inequality Problem

TL;DR

This paper introduces the Split Variational Inequality Problem (SVIP), a new framework for solving interconnected inverse problems via iterative algorithms in Hilbert spaces, with some cases being novel even in Euclidean spaces.

Contribution

It defines the SVIP, proposes iterative algorithms for solving it, and explores special cases, some of which are new in Euclidean space.

Findings

Proposed iterative algorithms for SVIP

Established convergence conditions in Hilbert spaces

Identified new special cases in Euclidean spaces

Abstract

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.