  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.
