Solving Linear Equations Over Maxmin-$\omega$ Systems

TL;DR

This paper introduces a novel method to solve maxmin-$$ linear systems, generalizing existing max-plus and min-plus systems, and reveals that these systems can have a finite number of solutions even when non-unique.

Contribution

It presents a new approach to solve maxmin-$$ linear systems by normalization and canonical matrix construction, extending methods from max-plus and min-plus systems.

Findings

Solutions can be efficiently identified using the principal order matrix.

Fully active solutions can be obtained directly from solution indices.

Maxmin-$$ systems can have a finite number of solutions in non-unique cases.

Abstract

Maxmin-$\omega$ dynamical systems were previously introduced as an ``all-in-one package'' that can yield a solely min-plus, a solely max-plus, or a max-min-plus dynamical system by varying a parameter $\omega\in(0,1]$. With such systems in mind, it is natural to introduce and consider maxmin-$\omega$ linear systems of equations of the type $A\otimes_{\omega} x=b$. However, to our knowledge, such maxmin-$\omega$ linear systems have not been studied before and in this paper we present an approach to solve them. We show that the problem can be simplified by performing normalization and then generating a ``canonical'' matrix which we call the principal order matrix. Instead of directly trying to find the solutions, we search the possible solution indices which can be identified using the principal order matrix and the parameter $\omega$. The fully active solutions are then immediately obtained from these solution indices. With the fully active solutions at hand, we then present the method to find other solutions by applying a relaxation, i.e., increasing or decreasing some components of fully active solutions. This approach can be seen as a generalization of an approach that could be applied to solve max-plus or min-plus linear systems. Our results also shed more light on an unusual feature of maxmin-$\omega$ linear systems, which, unlike in the usual linear algebra, can have a finite number of solutions in the case where their solution is non-unique.