Optimality Conditions for Rational Minimax Approximations: Bridging Ruttan's Criteria to Dual-Based Methods

TL;DR

This paper bridges Ruttan's optimality conditions for rational minimax approximations with dual-based computational methods, providing theoretical insights and practical criteria for ensuring global optimality in both discrete and continuum cases.

Contribution

It extends second-order optimality criteria, links Ruttan's conditions with the d-Lawson dual method, and shows how continuum problems can be approximated discretely for efficient computation.

Findings

Ruttan's sufficient condition becomes necessary with minimal extreme points.

Strong duality in d-Lawson ensures Ruttan's and Kolmogorov's criteria are met.

Continuum minimax approximants can be approximated discretely at boundary points.

Abstract

This paper presents a theoretical discussion on Ruttan's optimality conditions for rational minimax approximations in discrete and continuum settings, integrating analytical foundations with computational practice. We develop extended second-order optimality criteria for the discrete case, demonstrating that Ruttan's sufficient condition for global solutions [Ruttan, {Constr. Approx.}, 1 (1985), 287-296] becomes necessary when the number of extreme points is minimal. Our analysis further uncovers fundamental relationships between these conditions and the dual-based {d-Lawson} method [L.-H. Zhang et al., {Math. Comp.}, 94 (2025), 2457-2494], proving that strong duality in {d-Lawson} ensures simultaneous satisfaction of both Ruttan's and Kolmogorov's criteria. Additionally, we show that minimax approximants on a continuum satisfying Ruttan's sufficient global optimality can be captured through discrete minimax approximations at properly chosen boundary points, thereby enabling efficient computation of minimax approximants on a continuum using discrete methods.