Testing Surrogate-Based Optimization with the Fortified Branin-Hoo Extended to Four Dimensions

TL;DR

This study evaluates how fortifying the Branin-Hoo function affects surrogate-based optimization algorithms, revealing increased difficulty in higher dimensions and comparing Gaussian process and RBF methods in terms of efficiency and statistical robustness.

Contribution

It extends the analysis of fortified functions to four dimensions and compares the performance of EGO and RBF surrogate algorithms under these conditions.

Findings

Fortification increases optimization difficulty, especially in higher dimensions.

EGO requires fewer evaluations but is computationally expensive.

RBF is more cost-effective and provides better statistical data.

Abstract

Some popular functions used to test global optimization algorithms have multiple local optima, all with the same value, making them all global optima. It is easy to make them more challenging by fortifying them via adding a localized bump at the location of one of the optima. In previous work the authors illustrated this for the Branin-Hoo function and the popular differential evolution algorithm, showing that the fortified Branin-Hoo required an order of magnitude more function evaluations. This paper examines the effect of fortifying the Branin-Hoo function on surrogate-based optimization, which usually proceeds by adaptive sampling. Two algorithms are considered. The EGO algorithm, which is based on a Gaussian process (GP) and an algorithm based on radial basis functions (RBF). EGO is found to be more frugal in terms of the number of required function evaluations required to identify the correct basin, but it is expensive to run on a desktop, limiting the number of times the runs could be repeated to establish sound statistics on the number of required function evaluations. The RBF algorithm was cheaper to run, providing more sound statistics on performance. A four-dimensional version of the Branin-Hoo function was introduced in order to assess the effect of dimensionality. It was found that the difference between the ordinary function and the fortified one was much more pronounced for the four-dimensional function compared to the two dimensional one.