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

arXiv: https://arxiv.org/abs/2605.00855 · HTML: https://mlsanity.com/paper/2605.00855/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 54 references that are in the map.
- First posted on arXiv 21 Apr 2026.

## Builds on (most cited first, 16 of 16)

- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [Strong mixed-integer programming formulations for trained neural networks](https://mlsanity.com/paper/1811.08359/) (2018)
- [Benchmarking the Performance of Bayesian Optimization across Multiple Experimental Materials Science Domains](https://mlsanity.com/paper/2106.01309/) (2021)
- [Partition-based formulations for mixed-integer optimization of trained ReLU neural networks](https://mlsanity.com/paper/2102.04373/) (2021)
- [When Deep Learning Meets Polyhedral Theory: A Survey](https://mlsanity.com/paper/2305.00241/) (2023)
- [Local policy search with Bayesian optimization](https://mlsanity.com/paper/2106.11899/) (2021)
- [ReLU Networks as Surrogate Models in Mixed-Integer Linear Programs](https://mlsanity.com/paper/1907.03140/) (2019)
- [COBALT: COnstrained Bayesian optimizAtion of computationaLly expensive grey-box models exploiting derivaTive information](https://mlsanity.com/paper/2105.04114/) (2021)
- [Bayesian optimization as a flexible and efficient design framework for sustainable process systems](https://mlsanity.com/paper/2401.16373/) (2024)
- [Global Optimization of Gaussian processes](https://mlsanity.com/paper/2005.10902/) (2020)
- [Global Optimization of Gaussian Process Acquisition Functions Using a Piecewise-Linear Kernel Approximation](https://mlsanity.com/paper/2410.16893/) (2024)
- [Nonconvex piecewise linear functions: Advanced formulations and simple modeling tools](https://mlsanity.com/paper/1708.00050/) (2017)
- [NeST-BO: Fast Local Bayesian Optimization via Newton-Step Targeting of Gradient and Hessian Information](https://mlsanity.com/paper/2510.05516/) (2025)
- [Optimizing Posterior Samples for Bayesian Optimization via Rootfinding](https://mlsanity.com/paper/2410.22322/) (2024)
- [Deterministic Global Optimization of the Acquisition Function in Bayesian Optimization: To Do or Not To Do?](https://mlsanity.com/paper/2503.03625/) (2025)
- [Optimization with Trained Machine Learning Models Embedded](https://mlsanity.com/paper/2207.12722/) (2022)

## Abstract (arXiv)

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.