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A generalization of the relaxation-based matrix splitting iterative method for solving the system of generalized absolute value equations

Xuehua Li, Cairong Chen, Deren Han

TL;DR

The paper introduces GRMS, a momentum-accelerated generalization of RMS for solving $A x - B|x| - b = 0$, by employing a two-block reformulation with $|x|=Qy$ and a matrix splitting $A=M-N$, $Q=Q_1-Q_2$. It proves existence and uniqueness of the GAVEs solution under Cauchy convergence principles and provides a concrete convergence theorem with explicit bounds on norms, along with numerous corollaries that recover several existing schemes (e.g., RMS, NSOR, MFPI, FPI) as special cases. A set of comparison theorems between GRMS and competing methods (RMS, MGSOR, NSOR) is established, showing conditions under which GRMS converges faster. Numerical experiments on two AVEs demonstrate that GRMS outperforms a wide range of methods in iteration count and total CPU time, validating its practical effectiveness. The work delivers a flexible, theoretically grounded framework for solving GAVEs with potential extensions to singular $B$ cases and related momentum-accelerated schemes.

Abstract

By incorporating a new matrix splitting and the momentum acceleration into the relaxed-based matrix splitting (RMS) method \cite{soso2023}, a generalization of the RMS (GRMS) iterative method for solving the generalized absolute value equations (GAVEs) is proposed. On the one hand, unlike some existing methods, by using the Cauchy's convergence principle we give some sufficient conditions for the existence and uniqueness of the solution to GAVEs and prove that our method can converge to the unique solution of GAVEs. On the other hand, we obtain a few new and weaker convergence conditions for some existing methods. Moreover, we establish comparison theorems between GRMS method and some existing methods. Preliminary numerical experiments show that the proposed method is efficient.

A generalization of the relaxation-based matrix splitting iterative method for solving the system of generalized absolute value equations

TL;DR

The paper introduces GRMS, a momentum-accelerated generalization of RMS for solving , by employing a two-block reformulation with and a matrix splitting , . It proves existence and uniqueness of the GAVEs solution under Cauchy convergence principles and provides a concrete convergence theorem with explicit bounds on norms, along with numerous corollaries that recover several existing schemes (e.g., RMS, NSOR, MFPI, FPI) as special cases. A set of comparison theorems between GRMS and competing methods (RMS, MGSOR, NSOR) is established, showing conditions under which GRMS converges faster. Numerical experiments on two AVEs demonstrate that GRMS outperforms a wide range of methods in iteration count and total CPU time, validating its practical effectiveness. The work delivers a flexible, theoretically grounded framework for solving GAVEs with potential extensions to singular cases and related momentum-accelerated schemes.

Abstract

By incorporating a new matrix splitting and the momentum acceleration into the relaxed-based matrix splitting (RMS) method \cite{soso2023}, a generalization of the RMS (GRMS) iterative method for solving the generalized absolute value equations (GAVEs) is proposed. On the one hand, unlike some existing methods, by using the Cauchy's convergence principle we give some sufficient conditions for the existence and uniqueness of the solution to GAVEs and prove that our method can converge to the unique solution of GAVEs. On the other hand, we obtain a few new and weaker convergence conditions for some existing methods. Moreover, we establish comparison theorems between GRMS method and some existing methods. Preliminary numerical experiments show that the proposed method is efficient.
Paper Structure (9 sections, 24 theorems, 96 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 9 sections, 24 theorems, 96 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The GRMS iterative method
  3. Convergence analysis
  4. Comparison theorems
  5. GRMS vs. RMS
  6. GRMS vs. MGSOR
  7. GRMS vs. NSOR
  8. Numerical results
  9. Conclusions

Key Result

Theorem 1

Assume that $A,B,H\in \mathbb{R}^{n\times n}$, and $Q\in \mathbb{R}^{n\times n}$ is nonsingular. Let $A = M - N$ and $Q = Q_1 - Q_2$ with $M$ and $Q_1$ being nonsingular. Denote If and then GAVEs eq:gave has a unique solution $x^*$ for any $b\in \mathbb{R}^n$ and the sequence $\{(x^{(k)},y^{(k)})\}_{k=0}^{\infty}$ generated by the GRMS iteration eq:grmsconverges to $(x^*, y^*=Q^{-1}|x^*|)$.

Figures (3)

  • Figure 1: The curve of RES:=$\frac{\|Ax^{(k)} - B|x^{(k)}| -b\|_2}{\|b\|_2}$ and IT (the number of iteration) between GRMS and RMS iteration. The methods are terminated when ${\rm RES} \leq 10^{-9}$.
  • Figure 2: The curve of RES:=$\frac{\|Ax^{(k)} - |x^{(k)}| -b\|_2}{\|b\|_2}$ for Example \ref{['ex:mgsor']}. The methods are terminated when ${\rm RES} \leq 10^{-9}$.
  • Figure 3: The curve of RES:=$\frac{\|Ax^{(k)} - |x^{(k)}| -b\|_2}{\|b\|_2}$ and IT (the number of iteration) between GRMS and NSOR iteration. The experiment is terminated if RES $\leq 10^{-9}$.

Theorems & Definitions (46)

  • Theorem 1
  • proof
  • Remark 1
  • Corollary 1
  • proof
  • Corollary 2
  • Remark 2
  • Corollary 3
  • Remark 3
  • Corollary 4
  • ...and 36 more