The Split Common Null Point Problem

TL;DR

This paper introduces the Split Common Null Point Problem (SCNPP), a generalization of the Split Variational Inequality Problem, and proposes four iterative algorithms with convergence guarantees for solving it in Hilbert spaces.

Contribution

It defines the SCNPP for set-valued maximal monotone mappings and develops four new algorithms with proven convergence properties.

Findings

Four iterative algorithms for SCNPP with convergence guarantees

Weak convergence established for one algorithm, strong convergence for three

Generalizes previous split variational inequality problems

Abstract

We introduce and study the Split Common Null Point Problem (SCNPP) for set-valued maximal monotone mappings in Hilbert spaces. This problem generalizes our Split Variational Inequality Problem (SVIP) [Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numerical Algorithms 59 (2012), 301--323]. The SCNPP with only two set-valued mappings entails finding a zero of a maximal monotone mapping in one space, the image of which under a given bounded linear transformation is a zero of another maximal monotone mapping. We present four iterative algorithms that solve such problems in Hilbert spaces, and establish weak convergence for one and strong convergence for the other three.