Algebraic solutions of tropical optimization problems
N. Krivulin
TL;DR
The paper surveys tropical optimization problems defined over idempotent semifields, highlighting both linear and nonlinear objective functions and their solution techniques. It develops a unifying algebraic approach that introduces an auxiliary parameter representing the minimum objective value, transforming constrained problems into parametrized inequalities solvable via Kleene-star ($oldsymbol{A}^{oldsymbol{igstar}}$) representations. The authors compile and extend known results—covering Chebyshev approximation, span seminorms, linear-fractional programming, and spectral-radius problems—providing explicit, closed-form solutions and conditions for existence. A key contribution is a direct, complete solution to a new constrained problem, illustrating the practical power of the algebraic method for tropical optimization. Overall, the work advances the theory and toolbox for exact tropical optimization, with potential applications in scheduling, location analysis, transportation, and discrete event systems.
Abstract
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 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 functions.