AE regularity of interval matrices

TL;DR

This paper introduces the concept of AE regularity for interval matrices, characterizing conditions for guaranteed solvability of systems with quantified interval coefficients and exploring classes of matrices with this property.

Contribution

It defines AE regularity, analyzes its properties, characterizes certain matrix classes, and discusses computational aspects and open problems.

Findings

AE regularity ensures nonempty AE solution set for all right-hand sides

Some matrix classes are polynomially decidable for AE regularity

Open problems in computational complexity and characterization

Abstract

Consider a linear system of equations with interval coefficients, and each interval coefficient is associated with either a universal or an existential quantifier. The AE solution set and AE solvability of the system is defined by $\forall\exists$-quantification. Herein, we deal with the problem what properties must the coefficient matrix have in order that there is guaranteed an existence of an AE solution. Based on this motivation, we introduce a concept of AE regularity, which implies that the AE solution set is nonempty and the system is AE solvable for every right-hand side. We discuss characterization of AE regularity, and we also focus on various classes of matrices that are implicitly AE regular. Some of these classes are polynomially decidable, and therefore give an efficient way for checking AE regularity. We also state open problems related to computational complexity and characterization.