# A general mathematical framework for constrained mixed-variable blackbox optimization problems with meta and categorical variables

arXiv: https://arxiv.org/abs/2204.00886 · HTML: https://mlsanity.com/paper/2204.00886/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Apr 2022.

## Built on by (most cited first, 8 of 8)

- [SMT 2.0: A Surrogate Modeling Toolbox with a focus on Hierarchical and Mixed Variables Gaussian Processes](https://mlsanity.com/paper/2305.13998/) (2023)
- [High-dimensional mixed-categorical Gaussian processes with application to multidisciplinary design optimization for a green aircraft](https://mlsanity.com/paper/2311.06130/) (2023)
- [A distance for mixed-variable and hierarchical domains with meta variables](https://mlsanity.com/paper/2405.13073/) (2024)
- [Modeling Hierarchical Spaces: A Review and Unified Framework for Surrogate-Based Architecture Design](https://mlsanity.com/paper/2506.22621/) (2025)
- [Bayesian Optimization of a Lightweight and Accurate Neural Network for Aerodynamic Performance Prediction](https://mlsanity.com/paper/2503.19479/) (2025)
- [Surrogate-based optimization of system architectures subject to hidden constraints](https://mlsanity.com/paper/2504.08721/) (2025)
- [Global and Preference-based Optimization with Mixed Variables using Piecewise Affine Surrogates](https://mlsanity.com/paper/2302.04686/) (2023)
- [Transfer Learning in Bayesian Optimization for Aircraft Design](https://mlsanity.com/paper/2603.28999/) (2026)

## Abstract (arXiv)

A mathematical framework for modelling constrained mixed-variable optimization problems is presented in a blackbox optimization context. The framework introduces a new notation and allows solution strategies. The notation framework allows meta and categorical variables to be explicitly and efficiently modelled, which facilitates the solution of such problems. The new term meta variables is used to describe variables that influence which variables are acting or nonacting: meta variables may affect the number of variables and constraints. The flexibility of the solution strategies supports the main blackbox mixed-variable optimization approaches: direct search methods and surrogate-based methods (Bayesian optimization). The notation system and solution strategies are illustrated through an example of a hyperparameter optimization problem from the machine learning community.