A solution of a tropical linear vector equation

TL;DR

This paper addresses solving tropical linear vector equations by transforming them into tropical optimization problems, establishing conditions for solutions, and providing a general solution framework within idempotent mathematics.

Contribution

It introduces a novel approach that reduces tropical linear equations to optimization problems, with new existence, uniqueness conditions, and a general solution.

Findings

Established conditions for solution existence and uniqueness

Reduced the problem to a tropical optimization framework

Provided a general solution to the tropical linear vector equation

Abstract

A linear vector equation is considered defined in terms of idempotent mathematics. To solve the equation, we apply an approach that is based on the analysis of distances between vectors in idempotent vector spaces and reduces the solution of the equation to that of a tropical optimization problem. Based on the approach, existence and uniqueness conditions are established for the solution, and a general solution to the equation is given.