TY - JOUR

T1 - A multidimensional tropical optimization problem with a nonlinear objective function and linear constraints

AU - Krivulin, N.

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

KW - idempotent semifield

KW - multidimensional optimization problem

KW - non-linear objective function

KW - linear inequality constraints

KW - project scheduling

UR - https://arxiv.org/abs/1303.0542

U2 - 10.1080/02331934.2013.840624

DO - 10.1080/02331934.2013.840624

M3 - Article

VL - 64

SP - 1107

EP - 1129

JO - Optimization

JF - Optimization

SN - 0233-1934

IS - 5

ER -