An Approach Combining an Efficient and Global Evolutionary Algorithm with a Gradient-Based Method for Airfoil Design Problems

Abstract

ABSTRACT This paper presents a hybrid optimization approach using an evolutionary algorithm and a gradient-based method with an efficient weighting function for multi-objective problems (MoPs) to obtain deterministic solutions in real-world design. The proposed method starts with finding the approximate Pareto optimal set using the multi-objective evolutionary algorithm (MOEA). In this study, a non-dominated sorting genetic algorithm (NSGA-II) with multi-modal distribution crossover (MMDX) was applied to GA exploration as the initial data set. Then, the gradient-based method was used to find the deterministic non-dominated solution using the non-dominated solutions obtained using the MOEA. A quasi-Newton method was employed as a gradient-based method for unconstrained problems whereas sequential quadratic programming (SQP) was used for constrained problems. An efficient weighting function was proposed to define the weights for the scalarizing function. Test problems with and without constraints were considered for numerical experiments that compared the proposed method and the uniform weighting function, which was calculated based on the trade-off ratio conventionally used for translating MoPs to a single objective problem. The airfoil design problem was also solved as a real-world problem. This problem has two objectives: to maximize thickness and to minimize the drag force (Cd). Comparing the cover rate metrics (CR), the result of the proposed hybrid method was a uniform non-dominated solution that maintained the diversity of solutions compared with the conventional weighting function. In addition, the airfoil design solution demonstrated that this method can maintain higher solution diversity compared to the NSGA-II.