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

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.