A comparative study of multi-objective expected improvement for aerodynamic design

Multi-objective optimization in aerodynamics plays an important role in revealing trade-offs between conflicting objectives in order to discover important knowledge and insight for better future design. Of interest here is the use of Kriging surrogate models incorporated into a sequential Bayesian optimization (BO) strategy. In this paper, we studied four variants of multi-objective BO (MOBO) techniques that are based on expected improvement (EI), that is, Euclidean-based EI (EEI), expected hypervolume improvement (EHVI), ParEGO, and expected inverted penalty boundary intersection improvement (EIPBII) to understand their capabilities on handling multi-objective aerodynamic optimization problems. Numerical tests were performed on a set consisting of six generalized Schaffer problems (GSP), five low-fidelity, and one high-fidelity airfoil design problems. Results suggest that EHVI is the only method which consistently performed well on artificial and aerodynamic problems. EEI yields the worst performance and is not suitable to deal with various problem complexities. ParEGO, although it performs modestly on GSP, surprisingly works well on the low- and high-fidelity problems. On the other hand, EIPBII encounters the opposite case, where it is one of the best performer on GSP but yields modest performance on the aerodynamic problems. In light of the results, we suggest that EHVI is a highly potential MOBO method to be applied for multi-objective aerodynamic design optimization.