# Black-box Mixed-Variable Optimisation using a Surrogate Model that Satisfies Integer Constraints

arXiv: https://arxiv.org/abs/2006.04508 · HTML: https://mlsanity.com/paper/2006.04508/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jun 2020.

## Built on by (most cited first, 9 of 9)

- [Think Global and Act Local: Bayesian Optimisation over High-Dimensional Categorical and Mixed Search Spaces](https://mlsanity.com/paper/2102.07188/) (2021)
- [Bayesian Optimization over Discrete and Mixed Spaces via Probabilistic Reparameterization](https://mlsanity.com/paper/2210.10199/) (2022)
- [A mixed-categorical correlation kernel for Gaussian process](https://mlsanity.com/paper/2211.08262/) (2022)
- [EXPObench: Benchmarking Surrogate-based Optimisation Algorithms on Expensive Black-box Functions](https://mlsanity.com/paper/2106.04618/) (2021)
- [BARK: A Fully Bayesian Tree Kernel for Black-box Optimization](https://mlsanity.com/paper/2503.05574/) (2025)
- [Machine Learning for Combinatorial Optimisation of Partially-Specified Problems: Regret Minimisation as a Unifying Lens](https://mlsanity.com/paper/2205.10157/) (2022)
- [Mold into a Graph: Efficient Bayesian Optimization over Mixed-Spaces](https://mlsanity.com/paper/2202.00893/) (2022)
- [Weighted Euclidean Distance Matrices over Mixed Continuous and Categorical Inputs for Gaussian Process Models](https://mlsanity.com/paper/2503.02630/) (2025)
- [CatBOX: A Categorical-Continuous Bayesian Optimization with Spectral Mixture Kernels for Accelerated Catalysis Experiments](https://mlsanity.com/paper/2505.17393/) (2025)

## Abstract (arXiv)

A challenging problem in both engineering and computer science is that of minimising a function for which we have no mathematical formulation available, that is expensive to evaluate, and that contains continuous and integer variables, for example in automatic algorithm configuration. Surrogate-based algorithms are very suitable for this type of problem, but most existing techniques are designed with only continuous or only discrete variables in mind. Mixed-Variable ReLU-based Surrogate Modelling (MVRSM) is a surrogate-based algorithm that uses a linear combination of rectified linear units, defined in such a way that (local) optima satisfy the integer constraints. This method outperforms the state of the art on several synthetic benchmarks with up to 238 continuous and integer variables, and achieves competitive performance on two real-life benchmarks: XGBoost hyperparameter tuning and Electrostatic Precipitator optimisation.