  We consider optimization problems that are formulated and solved in the
framework of tropical mathematics. The problems consist in minimizing or
maximizing functionals defined on vectors of finite-dimensional semimodules
over idempotent semifields, and may involve constraints in the form of linear
equations and inequalities. The objective function can be either a linear
function or nonlinear function calculated by means of multiplicative conjugate
transposition of vectors. We start with an overview of known tropical
optimization problems and solution methods. Then, we formulate certain new
problems and present direct solutions to the problems in a closed compact
vector form suitable for further analysis and applications. For many problems,
the results obtained are complete solutions.
