Algebraic solution of tropical best approximation problems

TL;DR

This paper introduces a novel algebraic framework for solving tropical best approximation problems, providing explicit solutions and algorithms for polynomial and rational function approximation in tropical algebra, with applications to Chebyshev approximation.

Contribution

It develops a new approach to tropical approximation problems by reducing them to linear equations and provides explicit solutions and iterative algorithms for these problems.

Findings

Derived explicit solutions for tropical polynomial approximation.

Developed an iterative algorithm for rational approximation.

Applied methods to max-plus algebra for Chebyshev approximation.

Abstract

We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function, by tropical Puiseux polynomial and rational functions. A new solution approach is proposed, which involves the reduction of the problem of polynomial approximation to the best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation, and we evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form, by using an iterative alternating algorithm. To illustrate the new technique developed, we solve example approximation problems in terms of a real semifield, where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which corresponds to the best Chebyshev approximation by piecewise linear functions.