The Effect of Hessian Evaluations in the Global Optimization {\alpha}BB Method

TL;DR

This paper explores how symbolic computation of the Hessian matrix can improve the quality of convex underestimators in the {B} global optimization method by reducing conservativeness and enhancing accuracy.

Contribution

It demonstrates that symbolic manipulation of Hessian expressions can significantly improve underestimator quality in the {B} method.

Findings

Symbolic computation leads to less conservative underestimators.

Small manipulations in Hessian expressions can greatly affect underestimator quality.

Examples show improved bounds with symbolic Hessian evaluations.

Abstract

We consider convex underestimators that are used in the global optimization {\alpha}BB method and its variants. The method is based by augmenting the original nonconvex function by a relaxation term that is derived from an interval enclosure of the Hessian matrix. In this paper, we discuss the advantages of symbolic computation of the Hessian matrix. Symbolic computation often allows simplifications of the resulting expressions, which in turn means less conservative underestimators. We show by examples that even a small manipulation with the symbolic expressions, which can be processed automatically by computers, can have a large effect on the quality of underestimators.