Fast Calculation of Spectral Bounds for Hessian Matrices on Hyperrectangles

• Authors: M. Mönnigmann
• Venue: SIAM Journal on Matrix Analysis and Applications
• Year: 2011

This paper presents a fast method for the calculation of bounds on the spectra of Hessian matrix sets ${\cal H} \{\nabla^2 \varphi(x)|x \in S\}$ of nonlinear functions $\varphi : U\subset \mathbb{R}^n\to\mathbb{R}$ on hyperrectangles $S\subset U$. The new method differs from existing ones in that it deliberately does not use any interval matrices. Because interval matrices are never used, two interesting features result: (i) The new method requires only ${\cal O}(n) N(\varphi)$ operations (where $N(\varphi)$ denotes the number of operations necessary to evaluate $\varphi$ at a point in its domain), and (ii) for some (but not all) functions $\varphi$, the new method results in tighter eigenvalue bounds than the tight bounds for the interval Hessian matrix. This is surprising, since the fastest method for calculating the tight eigenvalue bounds for the interval Hessian requires ${\cal O}(2^n)$ operations. It is stressed, however, that it is easy to construct examples $\varphi$ for which the proposed method results in looser bounds than the tight bounds for the interval Hessian.