A multidimensional tropical optimization problem with a non-linear objective function and linear constraints

作者：Krivulin Nikolai^*

来源：Optimization, 2015, 64(5): 1107-1129.

DOI：10.1080/02331934.2013.840624

• idempotent semifield
• linear inequality constraints
• project scheduling
• multidimensional optimization problem
• non-linear objective function
• 65K10
• 90C48
• 90B35
• 15A80

We examine a multidimensional optimization problem in the tropical mathematics setting. The problem involves the minimization of a non-linear function defined on a finite-dimensional semimodule over an idempotent semifield subject to linear inequality constraints. We start with an overview of known tropical optimization problems with linear and non-linear 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 optimization 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 optimization problem under fairly general conditions. Numerical and graphical examples for two-dimensional problems are given to illustrate the obtained results.

• 出版日期2015-5-4