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A forward-reflected-anchored-backward splitting algorithm with double inertial effects for solving non-monotone inclusion problems

Nam Van Tran

Abstract

In this paper, we study inclusion problems where the involved operators may not be monotone in the classical sense. Specifically, we assume the operators to be generalized monotone, a weaker notion than classical monotonicity. This allows us to extend the applicability of our results to a broader class of operators. We apply the two-step inertial forward-reflected-anchored-backward splitting algorithm proposed in \cite{CHIN} to these non-monotone inclusion problems. We establish the strong convergence of the sequence generated by the algorithm and demonstrate its applicability to other optimization problems, including Constrained Optimization Problems, Mixed Variational Inequalities, and Variational Inequalities.

A forward-reflected-anchored-backward splitting algorithm with double inertial effects for solving non-monotone inclusion problems

Abstract

In this paper, we study inclusion problems where the involved operators may not be monotone in the classical sense. Specifically, we assume the operators to be generalized monotone, a weaker notion than classical monotonicity. This allows us to extend the applicability of our results to a broader class of operators. We apply the two-step inertial forward-reflected-anchored-backward splitting algorithm proposed in \cite{CHIN} to these non-monotone inclusion problems. We establish the strong convergence of the sequence generated by the algorithm and demonstrate its applicability to other optimization problems, including Constrained Optimization Problems, Mixed Variational Inequalities, and Variational Inequalities.

Paper Structure

This paper contains 10 sections, 9 theorems, 79 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some notions on convex analysis
  4. Monotone operators
  5. Algorithm and Convergence analysis
  6. Applications
  7. Application to COP
  8. Application to MVIP
  9. Application to VIP
  10. conclusion

Key Result

Lemma 2.1

CHIN Let $x,y,z \in \mathbb R^n$ and $a,b,\beta \in \mathbb{R}$. Then

Theorems & Definitions (18)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 8 more