An algorithm for best generalised rational approximation of continuous functions

TL;DR

This paper introduces an optimization algorithm for generalized rational approximation of continuous functions, leveraging the pseudo-convexity of the objective functions to improve efficiency in solving these approximation problems.

Contribution

The paper proves that the objective functions are pseudo-convex and develops efficient numerical methods for solving generalized rational approximation problems.

Findings

Objective functions are pseudo-convex in the sense of Penot and Quang.

Developed numerical methods are effective for a wide range of pseudo-convex functions.

Methods successfully tested on generalized rational approximation problems.

Abstract

The motivation of this paper is the development of an optimisation method for solving optimisation problems appearing in Chebyshev rational and generalised rational approximation problems, where the approximations are constructed as ratios of linear forms (linear combinations of basis functions). The coefficients of the linear forms are subject to optimisation and the basis functions are continuous function. It is known that the objective functions in generalised rational approximation problems are quasi-convex. In this paper we also prove a stronger result, the objective functions are pseudo-convex in the sense of Penot and Quang. Then we develop numerical methods, that are efficient for a wide range of pseudo-convex functions and test them on generalised rational approximation problems.