Optimisation with Intrinsic Dimension Reduction: A Ridge Informed Trust-Region Method

Abstract

High-dimensional design optimisation continues to present many challenges for engineers. Traditional model-based optimisation strategies can be difficult to implement for high-dimensional problems, as surrogate models often require sample sizes which are exponential in the number in number of variables. In this paper, we present an algorithm which combines ideas from ridge function approximations and trust-region methods to perform optimisation on high-dimensional functions with underlying low-dimensional structure. This approach allows us to efficiently explore the design space by focusing on the few directions in which the objective varies the most. By slowly increasing the problem dimension, this algorithm also can further explore regions of interest in detail. This allows for more accurate solutions than previous approaches which have used ridge functions during optimisation. We test this algorithm against a number of common model-based optimisation methods, and it is shown to be effective for a class of high-dimensional problems. In particular, we apply the method to a NACA 0012 aerofoil to obtain an optimal design with respect to drag. During this study, we demonstrate how ridge functions can be vital tools for design space exploration, and, with our algorithm, how they can be used for optimisation refinement.