Global Optimization of Gaussian processes

TL;DR

This paper introduces a new reduced-space formulation for deterministic global optimization with Gaussian processes, significantly improving computational efficiency and enabling scalable applications in Bayesian optimization and chance-constrained programming.

Contribution

It proposes a novel reduced-space approach with explicit Gaussian process models, envelopes of covariance functions, and tight relaxations, reducing computational time by orders of magnitude.

Findings

Achieves significant speedup over existing methods

Enables global optimization of Gaussian process-based models at larger scales

Demonstrates effectiveness in Bayesian optimization and chance-constrained problems

Abstract

Gaussian processes~(Kriging) are interpolating data-driven models that are frequently applied in various disciplines. Often, Gaussian processes are trained on datasets and are subsequently embedded as surrogate models in optimization problems. These optimization problems are nonconvex and global optimization is desired. However, previous literature observed computational burdens limiting deterministic global optimization to Gaussian processes trained on few data points. We propose a reduced-space formulation for deterministic global optimization with trained Gaussian processes embedded. For optimization, the branch-and-bound solver branches only on the degrees of freedom and McCormick relaxations are propagated through explicit Gaussian process models. The approach also leads to significantly smaller and computationally cheaper subproblems for lower and upper bounding. To further accelerate convergence, we derive envelopes of common covariance functions for GPs and tight relaxations of acquisition functions used in Bayesian optimization including expected improvement, probability of improvement, and lower confidence bound. In total, we reduce computational time by orders of magnitude compared to state-of-the-art methods, thus overcoming previous computational burdens. We demonstrate the performance and scaling of the proposed method and apply it to Bayesian optimization with global optimization of the acquisition function and chance-constrained programming. The Gaussian process models, acquisition functions, and training scripts are available open-source within the "MeLOn - Machine Learning Models for Optimization" toolbox~(https://git.rwth-aachen.de/avt.svt/public/MeLOn).