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

Lei-Hong Zhang

TL;DR

The paper tackles the challenge of characterizing and computing rational minimax approximants across discrete and continuum domains. It extends second-order optimality (Ruttan) and connects it to Kolmogorov first-order conditions, while embedding these insights in a dual-based framework via the $d$-Lawson iteration to solve the discrete problem efficiently. Key results include showing that strong duality ensures simultaneous satisfaction of both Kolmogorov and Ruttan criteria, proving that Ruttan's sufficient condition becomes necessary when the extremal set is minimal, and establishing a pathway to recover continuum minimax solutions from appropriately chosen boundary discretizations. The work thus provides a rigorous theoretical foundation for computing continuum rational minimax approximants through discrete methods and clarifies the roles of extremal structure and duality in global optimality.

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.

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

TL;DR

The paper tackles the challenge of characterizing and computing rational minimax approximants across discrete and continuum domains. It extends second-order optimality (Ruttan) and connects it to Kolmogorov first-order conditions, while embedding these insights in a dual-based framework via the -Lawson iteration to solve the discrete problem efficiently. Key results include showing that strong duality ensures simultaneous satisfaction of both Kolmogorov and Ruttan criteria, proving that Ruttan's sufficient condition becomes necessary when the extremal set is minimal, and establishing a pathway to recover continuum minimax solutions from appropriately chosen boundary discretizations. The work thus provides a rigorous theoretical foundation for computing continuum rational minimax approximants through discrete methods and clarifies the roles of extremal structure and duality in global optimality.

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.
Paper Structure (15 sections, 10 theorems, 66 equations, 1 algorithm)

This paper contains 15 sections, 10 theorems, 66 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Necessary optimality for local approximants
  3. The Kolmogorov conditions: results from first-order analysis
  4. The polynomial case
  5. The rational case
  6. Ruttan's necessary conditions: results from second-order analysis
  7. A summary of local optimality conditions
  8. Sufficient optimality for global approximants
  9. Ruttan's sufficient condition
  10. When Ruttan's sufficient condition becomes necessary
  11. Foundations for the d-Lawson iteration
  12. Strong duality and Ruttan's sufficient global optimality
  13. d-Lawson meets Kolmogorov's condition under strong duality
  14. From discrete boundary to continuum minimax solution
  15. Concluding remarks

Key Result

Lemma 2.1

If ${\cal F}\subset \mathbb{R}^n$, then every point of the convex hull ${\rm conv}({\cal F})=\left\{\sum_{j} \omega_j\pmb{y}_j :\sum_{j} \omega_j=1,~\omega_j\ge 0,~ \pmb{y}_j\in {\cal F}\right\}$ can be written as a convex linear combination of at most $n +1$ points of ${\cal F}$.

Theorems & Definitions (16)

  • Lemma 2.1: Carathéodory
  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2: Ruttan's primal formulation of second-order optimality conditions
  • proof
  • Theorem 2.3: Ruttan's dual formulation of second-order optimality conditions
  • proof
  • Corollary 2.1
  • proof
  • Theorem 3.1
  • ...and 6 more