On the Efficient Gerschgorin Inclusion Usage in the Global Optimization {\alpha}BB Method

TL;DR

This paper improves the {b} global optimization method by optimizing the Gerschgorin eigenvalue bounds through new heuristics for the scaling vector, leading to more efficient and effective bounds.

Contribution

It introduces two heuristics for computing the scaling vector in the {b} method and establishes necessary optimality conditions, enhancing bound computation efficiency.

Findings

The second heuristic satisfies all optimality conditions.

The proposed methods provide a cheap and efficient way to compute the scaling vector.

Improved eigenvalue bounds lead to better global optimization performance.

Abstract

In this paper, we revisit the {\alpha}BB method for solving global optimization problems. We investigate optimality of the scaling vector used in Gerschgorin's inclusion theorem to calculate bounds on the eigenvalues of the Hessian matrix. We propose two heuristics to compute good scaling vector d, and state three necessary optimality conditions for optimal d. Since the scaling vector calculated by the second presented method satisfies all three optimality conditions, it serves as a cheap but efficient solution.