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

TL;DR

This work tackles constrained optimization with mixed continuous, discrete, and categorical variables in high-cost simulations by introducing a Bayesian Quality-Diversity (BQD) framework. The method replaces objective, constraint, and feature evaluations with Gaussian Process surrogates using adapted kernels for mixed inputs and an infill strategy that balances quality and diversity. It also adapts MAP-Elites to handle mixed variables and constraints for the infill step, enabling efficient exploration under a limited budget. Experiments on analytical test functions and aerospace design problems demonstrate faster convergence and a richer diversity of high-quality solutions compared with MAP-Elites, with practical implications for early-stage engineering design.

Abstract

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.

Figures (30)

• Figure 1: General concepts of gradient-based (left) and population-based (right) optimization algorithms
• Figure 2: Quality-Diversity principle for a single continuous dimensional illustration ($x^c\in\mathbb{R}$) with two feature functions. The green niche (middle of the figure) is determined by the association of the two feature functions discretization (the yellow region of the feature 1 and the blue region of the feature 2, left of the figure). This niche defines a region of the design space (that can be a union of disjoint regions) in which the minimum value of the objective function has to be found (right of the figure).
• Figure 3: Proposed Bayesian QD algorithm to deal with mixed continuous, discrete and categorical variables and the presence of constraints
• Figure 4: Convergence curves (normalized QD score, the lower, the better) for the Rosenbrock problem with MAP-Elites and Bayesian QD algorithm with Gower and hypersphere kernels. For the ten repetitions, the curves correspond to the median whereas the upper and lower limits of the shade areas correspond to the $75^\text{th}$ and $25^\text{th}$ quantiles.
• Figure 5: Number of discovered niches (the higher, the better) for the Rosenbrock problem with MAP-Elites and Bayesian QD algorithm with Gower and hypersphere kernels. For the ten repetitions, the curves correspond to the median whereas the upper and lower limits of the shade corresponds to the $75^\text{th}$ and $25^\text{th}$ quantiles.