On the convergence and applications of the inertial-like method for null-point problems

TL;DR

This paper introduces two new inertial-like algorithms for solving split common null point problems, featuring a novel inertial structure and adaptive step-size selection, with demonstrated numerical effectiveness.

Contribution

The paper proposes inertial-like algorithms with a new inertial structure and adaptive step-size rules for improved convergence in null point problems.

Findings

Algorithms converge under the proposed inertial structure.

Numerical experiments show improved performance.

Step-size selection does not require prior operator norm knowledge.

Abstract

In this paper, we propose two novel inertial-like algorithms for solving the split common null point problem (SCNPP) with respect to set-valued maximal operators. The features of the presented algorithm are using new inertial structure (i.e, the design of the new inertial-like method does neither involve computation of the norm of the difference between $x_n$ and $x_{n-1}$ in advance, nor need to consider the special value of the inertial parameter $\theta_n$ to make the condition $\sum_{n=1}^\infty \alpha_n\|x_n-x_{n-1}\|^2<\infty$ valid) and the selection of the step-sizes does not need prior knowledge of operator norms. Numerical experiments are presented to illustrate the performance of the algorithms.