A PCA-assisted hybrid algorithm combining EAs and adjoint methods for CFD-based optimization

Hybrid optimization algorithms which combine Evolutionary Algorithms (EAs) and Gradient-Based (GB) methods o refine the most promising solutions, are valuable tools for use in engineering optimization. Several hybrid methods can be found in the literature; differences among them are associated with the criteria used to select individuals for refinement through the GB method and the feedback the EA gets from the latter. GB methods require the gradient of the objective functions with respect to the design variables. By employing the adjoint method in problems governed by partial differential equations, the cost of computing the gradient becomes independent of the number of design variables. For multi-objective optimization problems this paper is exclusively dealing with, the availability of the gradients of all objective functions is not enough. Hybrid algorithms require the computation of descent directions in the objective space capable of improving the current front of non-dominated solutions. Using the sum of weighted objectives as the new objective function is ineffective. In this paper, a method which refrains from using arbitrarily defined weights is proposed. The method is driven by data obtained from the Principal Component Analysis (PCA) of the objective function values of the elite individuals at each generation of the EA. The PCA, with computational cost that of the solution of an eigenproblem, identifies the direction in the objective space along which the current front of non-dominated solutions should be improved. This along with the gradients computed by the adjoint method are used by the GB method to refine selected individuals. The efficiency of the proposed hybrid algorithm is further improved by employing online trained surrogate models or metamodels and Kernel PCA within the EA-based search. The proposed method is demonstrated in aerodynamic shape optimization problems, using in-house Computational Fluid Dynamics software and its adjoint.