# Bayesian Quality-Diversity approaches for constrained optimization problems with mixed continuous, discrete and categorical variables

arXiv: https://arxiv.org/abs/2310.05955 · HTML: https://mlsanity.com/paper/2310.05955/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 60 references that are in the map.
- First posted on arXiv 11 Sep 2023.

## Builds on (most cited first, 8 of 8)

- [Illuminating search spaces by mapping elites](https://mlsanity.com/paper/1504.04909/) (2015)
- [Covariance Matrix Adaptation for the Rapid Illumination of Behavior Space](https://mlsanity.com/paper/1912.02400/) (2019)
- [Quality-Diversity Optimization: a novel branch of stochastic optimization](https://mlsanity.com/paper/2012.04322/) (2020)
- [A Latent Variable Approach to Gaussian Process Modeling with Qualitative and Quantitative Factors](https://mlsanity.com/paper/1806.07504/) (2018)
- [Deep Surrogate Assisted MAP-Elites for Automated Hearthstone Deckbuilding](https://mlsanity.com/paper/2112.03534/) (2021)
- [Efficient global optimization of constrained mixed variable problems](https://mlsanity.com/paper/1806.03975/) (2018)
- [Designing Air Flow with Surrogate-assisted Phenotypic Niching](https://mlsanity.com/paper/2105.04256/) (2021)
- [Revisiting Bayesian Optimization in the light of the COCO benchmark](https://mlsanity.com/paper/2103.16649/) (2021)

## Abstract (arXiv)

Complex system design problems, such as those involved in aerospace engineering, require the use of numerically costly simulation codes in order to predict the performance of the system to be designed. In this context, these codes are often embedded into an optimization process to provide the best design while satisfying the design constraints. Recently, new approaches, called Quality-Diversity, have been proposed in order to enhance the exploration of the design space and to provide a set of optimal diversified solutions with respect to some feature functions. These functions are interesting to assess trade-offs. Furthermore, complex design problems often involve mixed continuous, discrete, and categorical design variables allowing to take into account technological choices in the optimization problem. Existing Bayesian Quality-Diversity approaches suited for intensive high-fidelity simulations are not adapted to mixed variables constrained optimization problems. In order to overcome these limitations, a new Quality-Diversity methodology based on mixed variables Bayesian optimization strategy is proposed in the context of limited simulation budget. Using adapted covariance models and dedicated enrichment strategy for the Gaussian processes in Bayesian optimization, this approach allows to reduce the computational cost up to two orders of magnitude, with respect to classical Quality-Diversity approaches while dealing with discrete choices and the presence of constraints. The performance of the proposed method is assessed on a benchmark of analytical problems as well as on two aerospace system design problems highlighting its efficiency in terms of speed of convergence. The proposed approach provides valuable trade-offs for decision-markers for complex system design.