Many real-life design problems do not have access to derivative information. For instance, many design problems use open-source or commercial computational fluid dynamics (CFD) simulation codes to evaluate design performance. Although automatic differentiation and adjoints have become increasingly popular, acquiring derivative information from many of these simulation codes is often infeasible or intractable. For such problems, derivative-free optimization (DFO) methods offer a means of optimizing using only function evaluations. Unfortunately, many of these methods suffer from the curse of dimensionality — that is, as the problem dimension increases, these methods become less effective.

In order to combat the curse of dimensionality, two novel DFO methods are proposed — optimization by moving ridge functions (OMoRF) and constrained optimization by moving ridge functions (COMoRF). Both of these methods seek to reduce the effective problem dimension by combining trust region methodologies with dimension reduction techniques. In particular, surrogate models over a few linear combinations of the problem inputs are constructed and subsequently optimized in a trust region subproblem. These ridge functions require significantly fewer function evaluations for model construction, often allowing these methods to make substantially quicker progress than other DFO methods. Our proposed methods are the first to combine trust region and ridge function methodologies for unconstrained and constrained optimization without the use of any derivative information.

Our proposed methods are tested on a variety of optimization problems, including high-dimensional design optimization of centrifugal pump impellers. These impeller designs are parameterized by a total of 24 design parameters, which is significantly greater than what is often used for centrifugal pump impeller design. In this work, a total of three design optimization problems are performed: 1) efficiency maximization with constraints on head and wrap angle using steady-state CFD analysis, 2) efficiency maximization with constraints on head and wrap angle using transient CFD analysis and 3) average blade thickness maximization with constraints on efficiency, head and wrap angle using transient CFD analysis. These studies are some of the first truly high-dimensional centrifugal pump impeller design optimization studies which use fully transient CFD analysis. From these studies, it is shown that our proposed methods often substantially outperform other similar DFO methods — our methods ultimately achieving an impeller design which is nearly 1.5 points more efficient than and has an average blade thickness approximately 10 mm greater than a reference design.