A multidimensional tropical optimization problem with nonlinear objective function and linear constraints

TL;DR

This paper presents a comprehensive solution to a multidimensional tropical optimization problem involving nonlinear objectives and linear constraints, with applications demonstrated in project scheduling.

Contribution

It introduces a novel method that reduces the nonlinear tropical optimization problem to linear inequalities, providing a complete direct solution in a general setting.

Findings

Derived a necessary and sufficient condition for the inequality to hold

Provided a compact vector form solution for the optimization problem

Illustrated results with numerical and graphical examples

Abstract

We examine a multidimensional optimisation problem in the tropical mathematics setting. The problem involves the minimisation of a nonlinear function defined on a finite-dimensional semimodule over an idempotent semifield subject to linear inequality constraints. We start with an overview of known tropical optimisation problems with linear and nonlinear objective functions. A short introduction to tropical algebra is provided to offer a formal framework for solving the problem under study. As a preliminary result, a solution to a linear inequality with an arbitrary matrix is presented. We describe an example optimisation problem drawn from project scheduling and then offer a general representation of the problem. To solve the problem, we introduce an additional variable and reduce the problem to the solving of a linear inequality, in which the variable plays the role of a parameter. A necessary and sufficient condition for the inequality to hold is used to evaluate the parameter, whereas the solution to the inequality is considered a solution to the problem. Based on this approach, a complete direct solution in a compact vector form is derived for the optimisation problem under fairly general conditions. Numerical and graphical examples for two-dimensional problems are given to illustrate the obtained results.