Method of Alternating Projection for the Absolute Value Equation

TL;DR

This paper introduces a new method using alternating projections to solve the absolute value equation, providing convergence guarantees and handling cases where the matrix dimensions differ.

Contribution

It reformulates the absolute value equation as a feasibility problem and characterizes the fixed points, with proven linear convergence and applicability to non-square systems.

Findings

Proposes a MAP-based algorithm for absolute value equations.

Characterizes fixed points under nondegeneracy conditions.

Proves linear convergence of the proposed method.

Abstract

A novel approach for solving the general absolute value equation $Ax+B|x| = c$ where $A,B\in \mathbb{R}^{m\times n}$ and $c\in \mathbb{R}^m$ is presented. We reformulate the equation as a feasibility problem which we solve via the method of alternating projections (MAP). The fixed points set of the alternating projections map is characterized under nondegeneracy conditions on $A$ and $B$. Furthermore, we prove linear convergence of the algorithm. Unlike most of the existing approaches in the literature, the algorithm presented here is capable of handling problems with $m\neq n$, both theoretically and numerically.