Improved automatic computation of Hessian matrix spectral bounds

TL;DR

This paper introduces an efficient method for computing eigenvalue bounds of Hessian matrices for nonlinear functions, leveraging sparsity and avoiding interval matrices to reduce computational complexity.

Contribution

It improves existing spectral bound computation methods by exploiting sparsity and eliminating the need for interval matrices, enhancing efficiency and scalability.

Findings

The new method achieves linear complexity in the number of variables.

Exploiting sparsity significantly improves spectral bound computation.

The approach is faster and more scalable than traditional methods.

Abstract

This paper presents a fast and powerful method for the computation of eigenvalue bounds for Hessian matrices $\nabla^2 \varphi(x) $ of nonlinear functions $\varphi: U \subseteq R^n\rightarrow R$ on hyperrectangles $B \subset U$. The method is based on a recently proposed procedure for an efficient computation of spectral bounds using extended codelists. Both the previous approach and the one presented here substantially differ from established methods in that they do deliberately not use any interval matrices and thus result in a favorable numerical complexity of order $O(n)\,N(\varphi)$, where $N(\varphi)$ denotes the number of operations needed to evaluate $\varphi$ at a point in its domain. We improve the previous method by exploiting sparsity, which naturally arises in the underlying codelists.