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A convex dual problem for the rational minimax approximation and Lawson's iteration

Lei-Hong Zhang, Linyi Yang, Wei Hong Yang, Ya-Nan Zhang

TL;DR

This work addresses the global discrete rational minimax approximation in the complex plane by reformulating the nonconvex problem through a linearized primal and deriving a convex dual on the probability simplex. It proves that strong duality holds if and only if Ruttan's sufficient condition is satisfied, enabling the computation of the global minimax solution via the dual problem. A new Lawson-type iteration (d-Lawson) is introduced to solve the dual efficiently, with a filtering mechanism and stop rule tied to strong duality, and its monotonic progression in the dual objective is demonstrated numerically. The authors provide extensive real and complex-case experiments showing that d-Lawson is competitive with state-of-the-art methods like AAA, AAA-Lawson, RKFIT, and stabilized SK, while yielding verifiable minimax solutions under Ruttan's condition. This duality-based approach offers theoretical guarantees and a practical algorithmic route for reliable rational minimax approximation in applications requiring robust global optimality.

Abstract

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 approximations may not be minimax solutions. 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 approximation. Our numerical experiments demonstrate that this updated version of Lawson's iteration generally converges monotonically with respect to the objective function of the convex optimization. It is an effective competitive approach for computing the rational minimax approximation, compared to the highly efficient AAA, AAA-Lawson, and the stabilized Sanathanan-Koerner iteration.

A convex dual problem for the rational minimax approximation and Lawson's iteration

TL;DR

This work addresses the global discrete rational minimax approximation in the complex plane by reformulating the nonconvex problem through a linearized primal and deriving a convex dual on the probability simplex. It proves that strong duality holds if and only if Ruttan's sufficient condition is satisfied, enabling the computation of the global minimax solution via the dual problem. A new Lawson-type iteration (d-Lawson) is introduced to solve the dual efficiently, with a filtering mechanism and stop rule tied to strong duality, and its monotonic progression in the dual objective is demonstrated numerically. The authors provide extensive real and complex-case experiments showing that d-Lawson is competitive with state-of-the-art methods like AAA, AAA-Lawson, RKFIT, and stabilized SK, while yielding verifiable minimax solutions under Ruttan's condition. This duality-based approach offers theoretical guarantees and a practical algorithmic route for reliable rational minimax approximation in applications requiring robust global optimality.

Abstract

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 approximations may not be minimax solutions. 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 approximation. Our numerical experiments demonstrate that this updated version of Lawson's iteration generally converges monotonically with respect to the objective function of the convex optimization. It is an effective competitive approach for computing the rational minimax approximation, compared to the highly efficient AAA, AAA-Lawson, and the stabilized Sanathanan-Koerner iteration.
Paper Structure (22 sections, 10 theorems, 80 equations, 11 figures, 8 tables, 1 algorithm)

This paper contains 22 sections, 10 theorems, 80 equations, 11 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. A dual formulation of the rational minimax approximation
  3. A linearized reformulation
  4. A dual programming formulation
  5. Computation of $d_2(\pmb{w})$
  6. Optimality for the dual problem and computation of $d_2(\pmb{w})$
  7. More techniques for computing $d_2(\pmb{w})$
  8. Evaluation at new nodes
  9. Strong duality and Ruttan's sufficient condition
  10. The gradient and Hessian of $d_2(\pmb{w})$
  11. Lawson's iteration
  12. Lawson's iteration for the dual problem
  13. Other iteratively reweighted least-squares (IRLS) iterations
  14. Numerical results
  15. Numerical experiments on real cases
  16. ...and 7 more sections

Key Result

Theorem 2.1

Given $m\ge {n_1}+n_2+2$ distinct nodes ${\cal X}=\{x_j\}_{j=1}^m$ on $\Omega\subset\mathbb{C}$, suppose $\xi^*(x)=p^*(x)/q^*(x)\in \mathscr{R}_{(n_1,n_2)}$ is the unique solution eq:bestf which is irreducible and non-degenerate (i.e., $\upsilon(p^*,q^*)=0$ in eq:defect). Let Then for any $\alpha\ge 1$, the triple $(\eta_\infty^\alpha,p^*,q^*)$ is also the solution to the following problem

Figures (11)

  • Figure 1.1: Framework of handing the rational minimax approximation of \ref{['eq:bestf']}.
  • Figure 7.1: Error curves (blue) for type (4,4) approximant $\xi(x) \approx f_1(x)$ by the four methods.
  • Figure 7.2: The sequences of $\left\{e(\xi^{(k)})\right\}$ and $\left\{\sqrt{d_2(\pmb{w}^{(k)})}\right\}$ for d-Lawson (Algorithm \ref{['alg:Lawson']}) for approximating $|x|$ with type ${(4,4)}$.
  • Figure 7.3: Error curves of type ${(4,4)}$ to approximate $|x|$ with $\epsilon_w=0, 10^{-40},~10^{-30}$, respectively.
  • Figure 7.4: The maximum error $e(\xi)=\|\pmb{f}-\xi(\pmb{x})\|_\infty$ of the four methods with respect to various types $(n,n)$ in approximating $f_1(x)=|x|$ (left) and $f_2(x)=\sqrt{x}$ (right). Error curves with particular types $(n,n)$ are also plotted.
  • ...and 6 more figures

Theorems & Definitions (19)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.1
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Proposition 3.1
  • proof
  • ...and 9 more