An Overview of Absolute Value Equations: From Theory to Solution Methods and Challenges

TL;DR

This paper reviews the theory, solution methods, and challenges of absolute value equations, highlighting their NP-hardness, equivalence to linear complementarity problems, and discussing numerical approaches and open research issues.

Contribution

It provides a comprehensive overview of existence theorems, solution techniques, and open problems for absolute value equations, advancing understanding in this complex area.

Findings

NP-hardness of absolute value equations

Equivalence to linear complementarity problems

Discussion of numerical solution strategies

Abstract

This paper provides a thorough exploration of the absolute value equations $Ax-|x|=b$, a seemingly straightforward concept that has gained heightened attention in recent years. It is an NP-hard and nondifferentiable problem and equivalent with the standard linear complementarity problem. Offering a comprehensive review of existing literature, the study delves into theorems concerning the existence and nonexistence of solutions to the absolute value equations, along with numerical methods for effectively addressing this complex equation. Going beyond conventional approaches, the paper investigates strategies for obtaining solutions with minimal norms, techniques for correcting infeasible systems, and other pertinent topics. By pinpointing challenging issues and emphasizing open problems, this paper serves as a valuable guide for shaping the future research trajectory in this dynamic and multifaceted field.