A generalization of the Newton-based matrix splitting iteration method for generalized absolute value equations

TL;DR

This work tackles solving generalized absolute value equations of the form $A x - B|x| - c = 0$ by introducing the GNMS method, a generalization of Newton-based matrix-splitting schemes that leverages a matrix-splitting and a relaxation framework along with a $Qy$ transformation. The proposed two-stage iteration updates $y$ and then $x$, with a convergence analysis based on a contraction matrix $W$ whose spectral radius must satisfy $ ho(W)<1$, thereby ensuring convergence to the unique solution under mild conditions. The authors show that GNMS encompasses several existing methods (e.g., MN, NMS, RNMS and their relaxations) as special cases and provide weaker convergence conditions for these methods. Numerical experiments demonstrate GNMS’s superior efficiency in iteration count and computation time across tested scenarios, supporting its practical relevance for solving GAVEs in engineering and optimization contexts.

Abstract

A generalization of the Newton-based matrix splitting iteration method (GNMS) for solving the generalized absolute value equations (GAVEs) is proposed. Under mild conditions, the GNMS method converges to the unique solution of the GAVEs. Moreover, we can obtain a few weaker convergence conditions for some existing methods. Numerical results verify the effectiveness of the proposed method.

Theorems & Definitions (22)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Theorem 3.1
• proof
• Corollary 3.1
• proof