The extension of linear inequality method for generalised rational Chebyshev approximation to approximation by general quasilinear functions

TL;DR

This paper extends the linear inequality method for Chebyshev approximation to broader classes of quasiconvex and quasilinear functions, linking it with quasiconvex optimisation and offering new computational tools.

Contribution

It introduces a systematic extension of the linear inequality method to generalised rational and quasilinear Chebyshev approximation problems, connecting it with quasiconvex optimisation.

Findings

Establishes equivalence between linear inequality method and bisection in quasiconvex optimisation.

Extends the method to broader quasiconvex functions beyond linear inequalities.

Provides a more general approach for Chebyshev approximation involving convex feasibility problems.

Abstract

In this paper we demonstrate that a well known linear inequality method developed for rational Chebyshev approximation is equivalent to the application of the bisection method used in quasiconvex optimisation. Although this correspondence is not surprising, it naturally connects rational and generalised rational Chebyshev approximation problems with modern developments in the area of quasiconvex functions and therefore offers more theoretical and computational tools for solving this problem. %In particular, this observation leads a straightforward extension of the linear inequality method to generalised rational approximation in the sense of Loeb (ratio of two linear forms). The second important contribution of this paper is the extension of the linear inequality method to a broader class of Chebyshev approximation problems, where the corresponding objective functions remain quasiconvex. In this broader class of functions, the inequalities are no longer required to be linear: it is enough for each inequality to define a convex set and the computational challenge is in solving the corresponding convex feasibility problems. Therefore, we propose a more systematic and general approach for treating Chebyshev approximation problems. In particular, we are looking at the problems where the approximations are quasilinear functions with respect to their parameters, that are also the decision variables in the corresponding optimisation problems.