Algebraic solutions of tropical optimization problems

We consider multidimensional optimization problems, which are formulated and solved in terms of tropical mathematics. The problems are to minimize (maximize) a linear or nonlinear function defined on vectors of a finite-dimensional semimodule over an idempotent semifield, and may have constraints in the form of linear equations and inequalities. The aim of the paper is twofold: first to give a broad overview of known tropical optimization problems and solution methods, including recent results; and second, to derive a direct, complete solution to a new constrained optimization problem as an illustration of the algebraic approach recently proposed to solve tropical optimization problems with nonlinear objective function. Key-Words: idempotent semifield, tropical optimization problem, nonlinear objective function, linear inequality constraint, direct solution. MSC (2010): 65K05, 15A80, 90C48, 65K10