Algebraic solutions of tropical optimization problems

TL;DR

This paper provides an overview of tropical optimization problems and introduces a new algebraic method to solve constrained nonlinear optimization problems within tropical mathematics.

Contribution

It offers a comprehensive review of existing tropical optimization methods and presents a novel direct solution to a constrained nonlinear problem using algebraic techniques.

Findings

Broad overview of tropical optimization problems and methods

Derivation of a complete algebraic solution to a new constrained problem

Illustration of the algebraic approach for nonlinear objectives

Abstract

We consider multidimensional optimization problems, which are formulated and solved in terms of tropical mathematics. The problems are to minimize (maximize) a linear or nonlinear function defined on vectors over an idempotent semifield, and may have constraints in the form of linear equations and inequalities. The aim of the paper is twofold: first to give a broad overview of known tropical optimization problems and solution methods, including recent results; and second, to derive a direct, complete solution to a new constrained optimization problem as an illustration of the algebraic approach recently proposed to solve tropical optimization problems with nonlinear objective functions.