High-Dimensional Bayesian Optimisation with Large-Scale Constraints -- An Application to Aeroelastic Tailoring

TL;DR

This work tackles high-dimensional, constrained design optimisation by integrating Bayesian Optimisation with dimensionality reduction of constraint outputs. By projecting a large set of constraints onto a lower-dimensional subspace using PCA or kernel PCA, the authors train a smaller set of independent GPs (g < G) and apply constrained BO (SCBO) to achieve global search efficiently. The approach is validated on a 10D Ackley problem with 2 constraints, a 7D speed reducer with 11 constraints, and a practical aeroelastic tailoring problem, showing comparable or improved performance with substantial computational savings and scalability to large constraint sets. The results suggest broad applicability for complex MDO problems where thousands of constraints arise across loadcases or analyses, with future work exploring nonlinear projection methods and alternative scalable surrogates.

Abstract

Design optimisation potentially leads to lightweight aircraft structures with lower environmental impact. Due to the high number of design variables and constraints, these problems are ordinarily solved using gradient-based optimisation methods, leading to a local solution in the design space while the global space is neglected. Bayesian Optimisation is a promising path towards sample-efficient, global optimisation based on probabilistic surrogate models. While Bayesian optimisation methods have demonstrated their strength for problems with a low number of design variables, the scalability to high-dimensional problems while incorporating large-scale constraints is still lacking. Especially in aeroelastic tailoring where directional stiffness properties are embodied into the structural design of aircraft, to control aeroelastic deformations and to increase the aerodynamic and structural performance, the safe operation of the system needs to be ensured by involving constraints resulting from different analysis disciplines. Hence, a global design space search becomes even more challenging. The present study attempts to tackle the problem by using high-dimensional Bayesian Optimisation in combination with a dimensionality reduction approach to solve the optimisation problem occurring in aeroelastic tailoring, presenting a novel approach for high-dimensional problems with large-scale constraints. Experiments on well-known benchmark cases with black-box constraints show that the proposed approach can incorporate large-scale constraints.

Figures (8)

• Figure 1: Graphical interpretation of dimensionality reduction for constraints. On the left, PCA as a linear method is depicted, finding the lower dimensional subspace (blue arrow). On the right, the nonlinear extension, kPCA, is shown, first using a nonlinear kernel to map into the infinite dimensional space $\mathcal{F}$ and subsequently perform the standard PCA. The figure is inspired by scholkopf_nonlinear_1998.
• Figure 2: Beam representation of the wing structure
• Figure 3: Stiffness and thickness distributions (In-plane stiffness: black, out-of-plane stiffness: red), adopted from rajpal_dynamic_2021
• Figure 4: A comparison of the optimisation of the $10D$ Ackley function for SCBO and SCBO combined with PCA and kPCA. Only the objective values for feasible designs are plotted. All methods find a feasible optimum.
• Figure 5: $7D$ Speed reducer problem with $11$ black-box constraints from lemonge_2010. In (a), SCBO, SCBO-PCA and SCBO-kPCA are compared. In (b), the eigenvalues of the matrix $\textbf{C}$ are plotted