Efficient Computation of Spectral Bounds for Hessian Matrices on Hyperrectangles for Global Optimization

TL;DR

This paper evaluates and compares three methods for computing spectral bounds of Hessian matrices on hyperrectangles, focusing on their accuracy and computational efficiency in the context of global optimization problems.

Contribution

It introduces a new eigenvalue arithmetic method and compares it with established techniques using a large benchmark set, highlighting their relative performance.

Findings

Eigenvalue bounds vary in tightness and computational effort.

The new eigenvalue arithmetic method avoids interval Hessian computation.

Results inform method selection for global optimization tasks.

Abstract

We compare two established and a new method for the calculation of spectral bounds for Hessian matrices on hyperrectangles by applying them to a large collection of 1522 objective and constraint functions extracted from benchmark global optimization problems. Both the tightness of the spectral bounds and the computational effort are assessed. Specifically, we compare eigenvalue bounds obtained with the interval variant of Gershgorin's circle criterion [2,6], Hertz and Rohn's [7,16] method for tight bounds of interval matrices, and a recently proposed Hessian matrix eigenvalue arithmetic [12], which deliberately avoids the computation of interval Hessians.