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

TL;DR

This work addresses solving generalized absolute value equations $Ax-B|x|=b$ with non-square coefficient matrices by developing a flexible randomized iterative framework that uses sketching matrices. It establishes a comprehensive solvability theory, including a necessary-and-sufficient condition for unique solvability and a computable sufficiency criterion via a convex program to find a nonsingular matrix $M$, along with global error bounds. The core contribution is a versatile algorithmic framework that unifies and extends Picard, Kaczmarz, and block-Kaczmarz methods, proving almost-sure and linear convergence under explicit spectral conditions. Numerical experiments show that the proposed methods, especially randomized average block Kaczmarz, often outperform existing approaches and are effective for non-square problems and applications like ridge regression. The results broaden the applicability of randomized iterative methods to GAVE and offer practical guidelines for sketch choice and preconditioning.

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.

Figures (5)

• Figure 1: Comparison of RIMCS, RIMGS, RIMSRHT, and RABK. We set $p=10$. Figures depict the evolution of RSE with respect to the number of iterations (top) and the CPU time (bottom). The title of each plot indicates the values of $m,n,\kappa_A$, and $\kappa_B$.
• Figure 2: Figures depict the evolution of the number of full iterations (top) and the CPU time (bottom) with respect to the block size $p$. We fix $m=512$ and set $n$ to be $128,256$, and $512$. All computations are terminated once $\operatorname{RSE}<10^{-12}$.
• Figure 3: Figures depict the evolution of CPU time vs the increasing dimensions of the coefficient matrices. We have $m=n$ and the title of each plot indicates the values of $\kappa_A$ and $\kappa_B$.
• Figure 4: Comparison of SLA, MAP, and RABK with non-square coefficient matrices. Figures depict the CPU time (in seconds) vs increasing number of rows. The title of each plot indicates the values of $n,\kappa_A$, and $\kappa_B$.
• Figure 5: Comparison of PIM, GNM, MAP, and RABK for the asymmetric ridge regression. Figures depict the number of full iterations (left) and CPU time (right) vs increasing of $m$. The title of each plot indicates the values of $n$, $\bar{\lambda}$, and $\bar{\mu}$.

Theorems & Definitions (32)

• Lemma 2.1
• Lemma 2.2: zhang2011matrix
• Lemma 2.3: Lemma 2.3, lorenz2023minimal
• Theorem 3.1: Theorem 3.2, wu2021unique
• Theorem 3.2
• Lemma 3.3
• proof
• proof : Proof of Theorem \ref{['thm-solvability']}
• Theorem 3.4
• proof