Branch and Bound Algorithms for Maximizing Expected Improvement Functions

TL;DR

This paper introduces branch and bound algorithms to efficiently maximize expected improvement functions in sequential design for computer experiments, improving feature estimation accuracy over traditional methods.

Contribution

The paper develops novel branch and bound algorithms tailored for maximizing EI functions in specific feature estimation problems, outperforming existing optimization strategies.

Findings

Branch and bound algorithms outperform genetic algorithms in maximizing EI.

The proposed methods achieve higher accuracy in feature estimation with fewer runs.

Algorithms are effective for simultaneous estimation of multiple features.

Abstract

Deterministic computer simulations are often used as a replacement for complex physical experiments. Although less expensive than physical experimentation, computer codes can still be time-consuming to run. An effective strategy for exploring the response surface of the deterministic simulator is the use of an approximation to the computer code, such as a Gaussian process (GP) model, coupled with a sequential sampling strategy for choosing design points that can be used to build the GP model. The ultimate goal of such studies is often the estimation of specific features of interest of the simulator output, such as the maximum, minimum, or a level set (contour). Before approximating such features with the GP model, sufficient runs of the computer simulator must be completed. Sequential designs with an expected improvement (EI) function can yield good estimates of the features with a minimal number of runs. The challenge is that the expected improvement function itself is often multimodal and difficult to maximize. We develop branch and bound algorithms for efficiently maximizing the EI function in specific problems, including the simultaneous estimation of a minimum and a maximum, and in the estimation of a contour. These branch and bound algorithms outperform other optimization strategies such as genetic algorithms, and over a number of sequential design steps can lead to dramatically superior accuracy in estimation of features of interest.