有理最优化逼近的最优性条件：将Ruttan准则与基于对偶的方法连接起来
Optimality Conditions for Rational Minimax Approximations: Bridging Ruttan's Criteria to Dual-Based Methods
AI总结：
本文通过将Ruttan准则与基于对偶的方法结合，探讨了有理最优化逼近的最优性条件及连续情况下的计算方法。
摘要：
本文对离散和连续设置中有理最优化逼近的Ruttan最优性条件进行了理论讨论，结合了分析基础与计算实践。我们为离散情况开发了扩展的二阶最优性准则，证明了当极值点数量最小时，Ruttan为全局解的充分条件变为必要条件。我们的分析进一步揭示了这些条件与基于对偶的d-Lawson方法之间的基本关系，证明了d-Lawson中的强对偶性确保了Ruttan和Kolmogorov准则的同时满足。此外，我们还证明了在连续上满足Ruttan充分全局最优性的最优化逼近可通过在适当选择的边界点上进行离散最优化逼近来捕捉，从而使得通过离散方法高效计算连续上的最优化逼近成为可能。
英文摘要：
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.
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