An Efficient Spatial Branch-and-Bound Algorithm for Global Optimization of Gaussian Process Posterior Mean Functions

TL;DR

This paper introduces PALM-Mean, a novel spatial branch-and-bound algorithm that efficiently optimizes Gaussian process posterior means over hyperrectangles, improving scalability for large datasets.

Contribution

The paper presents a hybrid lower-bounding framework combining piecewise-analytic relaxations with analytical bounds, enabling scalable global optimization of Gaussian process means.

Findings

PALM-Mean provides valid lower bounds for the posterior mean.

The algorithm demonstrates improved scalability over existing solvers.

Computational results show effectiveness on synthetic and real-world problems.

Abstract

We study the deterministic global optimization of trained Gaussian process posterior mean functions over hyperrectangular domains. Although the posterior mean function has a compact closed-form representation, its global optimization is challenging because it remains nonlinear and nonconvex. Existing exact deterministic approaches become increasingly difficult to scale as the number of training data points grows, leading to approximation-based methods that improve tractability by optimizing a modified (inexact) objective. In this work, we propose PALM-Mean, a piecewise-analytic lower-bounding framework embedded in reduced-space spatial branch-and-bound. At each node, kernel terms that are locally important are replaced by a sign-aware piecewise-linear relaxation in an appropriate scalar distance variable, while the remaining terms are bounded analytically in closed form. We show this hybrid approach yields a valid lower bound for the posterior mean, while limiting the size of the branch-and-bound subproblems. We establish validity of the node lower bounds and $\varepsilon$-global convergence of the resulting algorithm. Computational results on synthetic benchmarks and real-world application problems show that PALM-Mean improves scalability relative to representative general-purpose deterministic global solvers, particularly as the number of training data points increases.