Nonlinear constrained optimization of Schur test functions

TL;DR

The work addresses obtaining refined upper bounds on the norms of transfer operators arising from Schur tests for integral kernels related to delay equations. It introduces an iterative nonlinear programming framework that reduces the full minimax problem to a sequence of smooth nonlinear programs by restricting the maximization to a growing finite set of reference points and solving with the SLSQP method. The method is demonstrated on kernels $K(\theta,s)$ from twofold additive compound delay operators, using neural-network parametrizations to model the Schur weights $(\mathfrak{p},\mathfrak{q})$ and testing Mackey–Glass-type scenarios; results show convergence with modest reference-point counts and sharp Schur bounds near $\omega=0$, with asymptotic guidance provided by the surrogate kernel $\bar{K}$. This provides a practical route to verify frequency inequalities for the global stability of delay equations and suggests extensions to systems of delay equations, along with data and code made available for reproducibility.

Abstract

We apply the iterative nonlinear programming method, previously proposed in our earlier work, to optimize Schur test functions and thereby provide refined upper bounds for the norms of integral operators. As an illustration, we derive such bounds for transfer operators associated with twofold additive compound operators that arise in the study of delay equations. This is related to the verification of frequency inequalities that guarantee the global stability of nonlinear delay equations through the generalized Bendixson criterion.

Figures (2)

• Figure 1: Dependence of the data versus iteration (horizontal) during the iterative nonlinear programming optimization of Schur test functions \ref{['EQ: SchurTestFunctionsN1N2']}-\ref{['EQ: SchurTestNNmodel']} applied to the kernel \ref{['EQ: ExampleResolventEquationsK2Formula']} with parameters \ref{['EQ: ExplicitResolventParamatersForTest']}, $\nu_{0} = 0.01$, and $\omega = 0$. (Left): Schur test estimate (red) and its optimized value over reference points (blue). (Right): Number of collected reference points (blue).
• Figure 2: (Left): Estimates for the norm of $T_{K}$ versus $\omega$ (horizontal) in the case of \ref{['EQ: ExplicitResolventParamatersForTest']} and $\nu_{0}=0.01$, namely, the $L_{2}$-norm of $K$ (red), the Schur test estimate (cyan), and the truncation norm (blue) for $N=50$ are shown. The horizontal lines pass through the threshold value $\Lambda^{-1}$ (orange), the $L_{2}$-norm of $\bar{K}$ (red), and the norm of $T_{\bar{K}}$ (olive) on the vertical axis. (Right): Plots of the optimized Schur test function $\mathfrak{p}$ for different $\omega$, colored using a blue-red colormap from $\omega = -20$ (blue) to $\omega = 20$ (red). See the repository for implementation details.