Deterministic global optimization with Gaussian processes embedded

Mathematical Programming Computation, vol. 13, pp. 553–581

Abstract

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 free variables 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—MachineLearning Models for Optimization” toolbox ( https://git.rwth-aachen.de/avt.svt/public/MeLOn ).

Authors 7

1. Artur M. Schweidtmann Aachen Process Systems Engineering (AVT.SVT)

   RWTH Aachen University

   Affiliation as printed

   Process Systems Engineering (AVT.SVT), RWTH Aachen University, Aachen, Germany

2. Dominik Bongartz Aachen Process Systems Engineering (AVT.SVT)

   RWTH Aachen University

   Affiliation as printed

   Process Systems Engineering (AVT.SVT), RWTH Aachen University, Aachen, Germany

3. Daniel Grothe Aachen Process Systems Engineering (AVT.SVT)

   RWTH Aachen University

   Affiliation as printed

   Process Systems Engineering (AVT.SVT), RWTH Aachen University, Aachen, Germany

4. Tim Kerkenhoff Aachen Process Systems Engineering (AVT.SVT)

   RWTH Aachen University

   Affiliation as printed

   Process Systems Engineering (AVT.SVT), RWTH Aachen University, Aachen, Germany

5. Xiaopeng Lin Aachen Process Systems Engineering (AVT.SVT)

   RWTH Aachen University

   Affiliation as printed

   Process Systems Engineering (AVT.SVT), RWTH Aachen University, Aachen, Germany

6. Jaromił Najman Aachen Process Systems Engineering (AVT.SVT)

   RWTH Aachen University

   Affiliation as printed

   Process Systems Engineering (AVT.SVT), RWTH Aachen University, Aachen, Germany

7. Alexander Mitsos corresponding Aachen Process Systems Engineering (AVT.SVT)

   Forschungszentrum Jülich · RWTH Aachen University

   Affiliation as printed

   Institute of Energy and Climate Research, Energy Systems Engineering (IEK-10), Forschungszentrum Jülich GmbH, Jülich, Germany

   JARA-CSD, Aachen, Germany

   Process Systems Engineering (AVT.SVT), RWTH Aachen University, Aachen, Germany