Properties of the solution set of absolute value equations and the related matrix classes

TL;DR

This paper analyzes the mathematical properties of the solution set of absolute value equations (AVEs), exploring topological features, solvability conditions, and the complexity of related matrix classes, providing new insights into their structure.

Contribution

It offers a comprehensive analysis of AVE solution set properties, characterizes associated matrix classes, and examines computational complexity, including polynomial recognition and NP-hardness results.

Findings

Solution set may be convex, bounded, or finite.

Recognition of certain matrix classes is polynomially feasible.

Some matrix class recognition problems are NP-hard.

Abstract

The absolute value equations (AVE) problem is an algebraic problem of solving Ax+|x|=b. So far, most of the research focused on methods for solving AVEs, but we address the problem itself by analysing properties of AVE and the corresponding solution set. In particular, we investigate topological properties of the solution set, such as convexity, boundedness, connectedness, or whether it consists of finitely many solutions. Further, we address problems related to nonnegativity of solutions such as solvability or unique solvability. AVE can be formulated by means of different optimization problems, and in this regard we are interested in how the solutions of AVE are related with optima, Karush-Kuhn-Tucker points and feasible solutions of these optimization problems. We characterize the matrix classes associated with the above mentioned properties and inspect the computational complexity of the recognition problem; some of the classes are polynomially recognizable, but some others are proved to be NP-hard. For the intractable cases, we propose various sufficient conditions. We also post new challenging problems that raised during the investigation of the problem.