Research output: Contribution to journal › Article › peer-review
A multidimensional tropical optimization problem with a nonlinear objective function and linear constraints. / Krivulin, N.
In: Optimization, Vol. 64, No. 5, 2015, p. 1107-1129.Research output: Contribution to journal › Article › peer-review
}
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 -
ID: 3987572