Comments on finite termination of the generalized Newton method for absolute value equations

TL;DR

This paper analyzes the generalized Newton method for the absolute value equation $A x - |x| = b$ and proves that finite termination occurs whenever the method converges. A key result is that if $\rho(|A^{-1}|) < 1/3$, GNM converges to the unique solution for any $b$ and any starting point $x^0$, with a contraction bound driven by $B = (I - |A^{-1}|)^{-1} |A^{-1}|$. The authors also derive sharp results for AVEs associated with $M$-matrices: when $A-I$ is a nonsingular $M$-matrix, convergence occurs within $n+2$ iterations; when $A-I$ is an irreducible singular $M$-matrix, various cases of $v^T b$ yield either a unique exact solution or a uniquely identifiable zero-containing solution, found in at most $n+1$ iterations. These findings extend convergence theory, provide explicit termination guarantees, and offer practical criteria for applying GNM to AVEs and related LCP reformulations.

Abstract

We consider the generalized Newton method (GNM) for the absolute value equation (AVE) $Ax-|x|=b$. The method has finite termination property whenever it is convergent, no matter whether the AVE has a unique solution. We prove that GNM is convergent whenever $ρ(|A^{-1}|)<1/3$. We also present new results for the case where $A-I$ is a nonsingular $M$-matrix or an irreducible singular $M$-matrix. When $A-I$ is an irreducible singular $M$-matrix, the AVE may have infinitely many solutions. In this case, we show that GNM always terminates with a uniquely identifiable solution, as long as the initial guess has at least one nonpositive component.

Theorems & Definitions (27)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• proof
• Proposition 7
• proof
• Example 1