Sequential parameter optimisation of evolutionary algorithms for airfoil design

More and more complex optimisation techniques play an increasing role in todays industry. Different techniques like gradient based methods or evolutionary search techniques are coupled (hybridisation, memetic algorithms [1]), enhanced by methods to fasten objective function evaluations (fitness approximation, metamodel assisted optimisation[2, 3]), or applied to more complex tasks with more than one objective function (multi-objective optimisation [4, 5]). Each of these enhanced techniques is able to improve optimisation results. Utilising not only one of them promises to further meliorate results, what is pushed by industrial needs. The drawback of such highly sophisticated methods and techniques is the growing number of parameters. Due to possible complex interactions, these parameters must by handled with care. A wrong parameter setting may lead to unwanted and bad optimisation results while the right parameter setting for the same algorithm-application combination may lead to extremely good results. This means, that the setting of parameters plays a major role in design optimisation. This article describes the sequential parameter optimization (SPO) framework [6, 7]. SPO has been succesfully applied to optimisation problems in the following domains: Machine engineering, Aerospace industry, Elevator group control, Algorithm engineering, Graph drawing, Algorithmic chemistry, Technical Thermodynamics, Agri-environmental policy-switchings, vehicle routing, and bioinformatics. Experimental setup The new technique to optimise parameter settings it applied to evolutionary (multi-objective) optimisation algorithms on airfoil design optimisation tasks. First, an older two-dimensional NACA-redesign testcase from some European research project is considered. It is described in more detail by Naujoks et.al. [8]. The hypervolume or S-metric is computed to measure the quality of the received Pareto-fronts. This quality indicator measures the covered space of the Pareto-front related to a reference point that is dominated by all solutions of the computed optimisation runs. The S-metric was incorporated for selection in recently presented EMO algorithms, e.g., the S-Metric Selection EMOA (SMA-EMOA)[9]. This algorithm features a (μ + 1)-selection scheme, where variation operators generate one new individual and the individual providing the least contribution to the hypervolume of the worst ranked front of the population is discarded in each generation. Obviously, the population size μ the variation operator are crucial parameters of the methods that are analysed here. The variation operators incorporated in the analysis are: • SBX crossover and polynomial mutation proposed by Deb [4] for multi-objective optimisation tasks (abbreviated “Deb” as well) and • Discrete and intermediate recombination on object parameters and step sizes, respectively as well as mutation featuring n (number of object parameters) step sizes like commonly used in evolution strategies [10] (abbreviated “ES” accordingly). For the accomplished analysis, we consider an optimisation run as an experiment. Tools from statistical design and analysis of experiments can be applied to perform and analyse optimisation runs. SPO combines methods from classical Design of Experiments (DOE) and modern approaches such as Classification and Regression Trees (CART) and Design and Analysis of Computer Experiments (DACE). Bartz-Beielstein [7] provides a comprehensive introduction. An SPO-toolbox is freely available (http://ls11-www.cs.unidortmund.de/people/tom/ExperimentalResearch.html). Sequential parameter optimisation tries to discover interesting features in the data. A closer look at the data is already sufficient in many situations, no high-level statistics are necessary if the data are ”well-prepared.” We discuss some ”datascopes” from EDA (explorative data analysis) first. EDA tools are useful to screen out worse configurations. High-level tools, which rely on complex regression models, can be used in the second phase. Here we can mention DACE stochastic process models.