Computing the discrete rational minimax approximation in the complex plane is challenging. Apart from Ruttan's sufficient condition, there are few other sufficient conditions for global optimality. The state-of-the-art rational approximation algorithms, such as the adaptive Antoulas-Anderson (AAA), AAA-Lawson, and the rational Krylov fitting (RKFIT) method, perform highly efficiently, but the computed rational approximants may be near-best. In this paper, we propose a convex programming approach, the solution of which is guaranteed to be the rational minimax approximation under Ruttan's sufficient condition. Furthermore, we present a new version of Lawson's iteration for solving this convex programming problem. The computed solution can be easily verified as the rational minimax approximant. Our numerical experiments demonstrate that this updated version of Lawson's iteration generally converges monotonically with respect to the objective function of the convex programming. It is an effective competitive approach for the rational minimax problem, compared to the highly efficient AAA, AAA-Lawson, and the stabilized Sanathanan-Koerner iteration.

翻译：在复平面中计算离散有理极小极大逼近具有挑战性。除Ruttan充分条件外，全局最优性的其他充分条件寥寥无几。当前最先进的有理逼近算法，如自适应Antoulas-Anderson（AAA）方法、AAA-Lawson方法以及有理Krylov拟合（RKFIT）方法，虽性能高效，但计算得到的有理逼近可能仅为近最优解。本文提出一种凸规划方法，其解在Ruttan充分条件下保证为有理极小极大逼近。此外，我们提出求解该凸规划问题的新版Lawson迭代算法。计算解可轻松验证为有理极小极大逼近。数值实验表明，该更新版Lawson迭代通常关于凸规划目标函数单调收敛。与高效的AAA、AAA-Lawson及稳定化Sanathanan-Koerner迭代相比，该方法对有理极小极大问题具有有效竞争力。