Efficient Calculation of Bounds on Spectra of Hessian Matrices

• Authors: M. Mönnigmann
• Venue: SIAM Journal on Scientific Computing
• Year: 2008

We introduce a new method for the calculation of bounds on eigenvalues of Hessian matrices $\nabla^2 \varphi(x)$ of twice continuously differentiable functions $\varphi:U\subseteq\mathbb{R}^n\rightarrow \mathbb{R}$. The computational complexity of the new approach is shown to be of order ${\cal O}(n)\,N(\varphi)$ where $N(\varphi)$ is the number of operations necessary to evaluate the function $\varphi(x)$ at a point in its domain. This result is surprising, since the complexity of the calculation of the Hessian itself is of order ${\cal O}(n^2)\,N(\varphi)$ if the same method is used as in the proposed eigenvalue bounding approach. The favorable complexity of the new approach results because the eigenvalue bounds can be found without ever calculating the Hessian matrix.