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

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. Key-Words: idempotent semifield, multidimensional optimisation problem, nonlinear objective function, linear inequality constraints, project scheduling. MSC (2010): 65K10, 15A80, 90C48, 90B35 Faculty of Mathematics and Mechanics, Saint Petersburg State University, 28 Universitetsky Ave., Saint Petersburg, 198504, Russia, [email protected]. The work was supported in part by the Russian Foundation for Humanities under Grant #13-02-00338.