Improved automatic computation of Hessian matrix spectral bounds

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.