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A generalization of the Gauss-Seidel iteration method for generalized absolute value equations

Tingting Luo, Jiayu Liu, Cairong Chen, Linjie Chen, Changfeng Ma

TL;DR

The paper addresses solving generalized absolute value equations $A x - B|x| = b$, including cases where $B$ may be singular. It proposes a parameter-free generalization of the Gauss-Seidel method, deriving a convergent fixed-point iteration based on splitting $A$ and $B$ into diagonal and triangular parts. Under mild conditions, including $D_A>|D_B|$ and a contraction bound, the method is proven to converge to the unique solution when it exists. Numerical experiments on problems arising from the LCP and with singular $B$ demonstrate that GGS is competitive in CPU time and robust to parameter choices, outperforming several parameter-tuned alternatives. The study extends prior results by Edalatpour et al. and broadens applicability to general GAVEs.

Abstract

A parameter-free method, namely the generalization of the Gauss-Seidel (GGS) method, is developed to solve generalized absolute value equations. Convergence of the proposed method is analyzed. Numerical results are given to demonstrate the effectiveness and efficiency of the GGS method. Some results in the recent work of Edalatpour et al. \cite{edhs2017} are extended.

A generalization of the Gauss-Seidel iteration method for generalized absolute value equations

TL;DR

The paper addresses solving generalized absolute value equations , including cases where may be singular. It proposes a parameter-free generalization of the Gauss-Seidel method, deriving a convergent fixed-point iteration based on splitting and into diagonal and triangular parts. Under mild conditions, including and a contraction bound, the method is proven to converge to the unique solution when it exists. Numerical experiments on problems arising from the LCP and with singular demonstrate that GGS is competitive in CPU time and robust to parameter choices, outperforming several parameter-tuned alternatives. The study extends prior results by Edalatpour et al. and broadens applicability to general GAVEs.

Abstract

A parameter-free method, namely the generalization of the Gauss-Seidel (GGS) method, is developed to solve generalized absolute value equations. Convergence of the proposed method is analyzed. Numerical results are given to demonstrate the effectiveness and efficiency of the GGS method. Some results in the recent work of Edalatpour et al. \cite{edhs2017} are extended.
Paper Structure (7 sections, 11 theorems, 55 equations, 3 figures, 3 tables, 3 algorithms)

This paper contains 7 sections, 11 theorems, 55 equations, 3 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The GGS method for GAVEs
  4. Numerical results
  5. GAVEs arising from LCP
  6. GAVEs with $B$ being singular
  7. Conclusion

Key Result

Lemma 2.1

Let $A\in \mathbb{R}^{n\times n}$ be an $M$-matrix. If $A=M-N$ is an $M$-splitting of $A$, then $\rho(M^{-1}N)<1$.

Figures (3)

  • Figure 1: Parametric influence on GGS and AMGS in Example \ref{['exam:5.1']}
  • Figure 2: Parametric influence on GGS and AMGS in Example \ref{['exam:5.2']}
  • Figure 3: Relationship between $m$ and inf_ norm in Example \ref{['exam:5.3']}

Theorems & Definitions (20)

  • Lemma 2.1: e.g., zhyi2013
  • Lemma 2.2: varg1962
  • Lemma 2.3: e.g., saad2003
  • Lemma 2.4: e.g., saad2003
  • Lemma 2.5: saad2003
  • Lemma 2.6: frma1989
  • Lemma 2.7: saad2003
  • Lemma 2.8: edhs2017
  • Lemma 3.1
  • proof
  • ...and 10 more