# Complete algebraic solution of multidimensional optimization problems in tropical semifield

## TL;DR

This paper provides a complete algebraic method for solving multidimensional optimization problems in tropical mathematics, including reducible cases, with a new solution representation and an efficient backtracking approach.

## Contribution

It introduces a comprehensive algebraic solution for reducible matrices in tropical optimization, extending known results and providing a closed-form characterization of all solutions.

## Key findings

- Derived the minimum value and solution set for reducible matrices.
- Represented solutions as families of subsets using matrix sparsification.
- Developed a backtracking procedure to efficiently generate solution subsets.

## Abstract

We consider multidimensional optimization problems that are formulated in the framework of tropical mathematics to minimize functions defined on vectors over a tropical semifield (a semiring with idempotent addition and invertible multiplication). The functions, given by a matrix and calculated through multiplicative conjugate transposition, are nonlinear in the tropical mathematics sense. We start with known results on the solution of the problems with irreducible matrices. To solve the problems in the case of arbitrary (reducible) matrices, we first derive the minimum value of the objective function, and find a set of solutions. We show that all solutions of the problem satisfy a system of vector inequalities, and then use these inequalities to establish characteristic properties of the solution set. Furthermore, all solutions of the problem are represented as a family of subsets, each defined by a matrix that is obtained by using a matrix sparsification technique. We describe a backtracking procedure that allows one to reduce the brute-force generation of sparsified matrices by skipping those, which cannot provide solutions, and thus offers an economical way to obtain all subsets in the family. Finally, the characteristic properties of the solution set are used to provide complete solutions in a closed form. We illustrate the results obtained with simple numerical examples.