Randomized iterative methods for generalized absolute value equations: Solvability and error bounds

TL;DR

This paper develops a comprehensive framework for applying randomized iterative methods to solve generalized absolute value equations with non-square matrices, establishing solvability conditions, error bounds, and convergence guarantees.

Contribution

It introduces a novel randomized iterative framework for GAVE with non-square matrices, including convergence analysis and numerical validation.

Findings

Established solvability conditions for GAVE

Proposed a flexible randomized algorithmic framework

Proved almost sure and linear convergence rates

Abstract

Randomized iterative methods, such as the Kaczmarz method and its variants, have gained growing attention due to their simplicity and efficiency in solving large-scale linear systems. Meanwhile, absolute value equations (AVE) have attracted increasing interest due to their connection with the linear complementarity problem. In this paper, we investigate the application of randomized iterative methods to generalized AVE (GAVE). Our approach differs from most existing works in that we tackle GAVE with non-square coefficient matrices. We establish more comprehensive sufficient and necessary conditions for characterizing the solvability of GAVE and propose precise error bound conditions. Furthermore, we introduce a flexible and efficient randomized iterative algorithmic framework for solving GAVE, which employs randomized sketching matrices drawn from user-specified distributions. This framework is capable of encompassing many well-known methods, including the Picard iteration method and the randomized Kaczmarz method. Leveraging our findings on solvability and error bounds, we establish both almost sure convergence and linear convergence rates for this versatile algorithmic framework. Finally, we present numerical examples to illustrate the advantages of the new algorithms.