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Several self-adaptive inertial projection algorithms for solving split variational inclusion problems

Zheng Zhou, Bing Tan, Songxiao Li

TL;DR

In inertial hybrid and shrinking projection algorithms are proposed under the effect of a self-adaptive stepsize which does not require information of the norms of the given operators.

Abstract

This paper is to analyze the approximation solution of a split variational inclusion problem in the framework of infinite dimensional Hilbert spaces. For this purpose, several inertial hybrid and shrinking projection algorithms are proposed under the effect of self-adaptive stepsizes which does not require information of the norms of the given operators. Some strong convergence properties of the proposed algorithms are obtained under mild constraints. Finally, an experimental application is given to illustrate the performances of proposed methods by comparing existing results.

Several self-adaptive inertial projection algorithms for solving split variational inclusion problems

TL;DR

In inertial hybrid and shrinking projection algorithms are proposed under the effect of a self-adaptive stepsize which does not require information of the norms of the given operators.

Abstract

This paper is to analyze the approximation solution of a split variational inclusion problem in the framework of infinite dimensional Hilbert spaces. For this purpose, several inertial hybrid and shrinking projection algorithms are proposed under the effect of self-adaptive stepsizes which does not require information of the norms of the given operators. Some strong convergence properties of the proposed algorithms are obtained under mild constraints. Finally, an experimental application is given to illustrate the performances of proposed methods by comparing existing results.

Paper Structure

This paper contains 12 sections, 10 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. State of problem and preliminaries
  3. Split variational inclusion problem
  4. Preliminaries
  5. Several self-adaptive inertial hybrid and shrinking projection algorithms
  6. The strong convergence of inertial hybrid projection algorithm
  7. The strong convergence of inertial shrinking projection algorithms
  8. Theoretical applications
  9. Split variational inequality problem
  10. Split saddle point problem
  11. Split minimization problem
  12. Numerical example

Key Result

lemma thmcounterlemma

marino2004convergencetakahashi2000 The resolvent mapping $J_\beta^{B}$ of a maximal monotone mapping $B$ with $\beta>0$ is defined as $J_\beta^{B}(x)=(I+\beta B)^{-1}(x), \forall x\in H$. The following properties associated with $J_\beta^{B}$ hold.

Theorems & Definitions (18)

  • definition thmcounterdefinition
  • remark thmcounterremark
  • lemma thmcounterlemma
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof
  • theorem 1
  • proof
  • ...and 8 more