Markus Olhofer

A framework for evolutionary optimization with approximate fitness functions

It is not unusual that an approximate model is needed for fitness evaluation in evolutionary computation. In this case, the convergence properties of the evolutionary algorithm are unclear due to the approximation error of the model. In this paper, extensive empirical studies are carried out to investigate the convergence properties of an evolution strategy using an approximate fitness function on two benchmark problems. It is found that incorrect convergence will occur if the approximate model has false optima. To address this problem, individual- and generation-based evolution control are introduced and the resulting effects on the convergence properties are presented. A framework for managing approximate models in generation-based evolution control is proposed. This framework is well suited for parallel evolutionary optimization, which is able to guarantee the correct convergence of the evolutionary algorithm, as well as to reduce the computation cost as much as possible. Control of the evolution and updating of the approximate models are based on the estimated fidelity of the approximate model. Numerical results are presented for three test problems and for an aerodynamic design example.

A Reference Vector Guided Evolutionary Algorithm for Many-Objective Optimization

In evolutionary multiobjective optimization, maintaining a good balance between convergence and diversity is particularly crucial to the performance of the evolutionary algorithms (EAs). In addition, it becomes increasingly important to incorporate user preferences because it will be less likely to achieve a representative subset of the Pareto-optimal solutions using a limited population size as the number of objectives increases. This paper proposes a reference vector-guided EA for many-objective optimization. The reference vectors can be used not only to decompose the original multiobjective optimization problem into a number of single-objective subproblems, but also to elucidate user preferences to target a preferred subset of the whole Pareto front (PF). In the proposed algorithm, a scalarization approach, termed angle-penalized distance, is adopted to balance convergence and diversity of the solutions in the high-dimensional objective space. An adaptation strategy is proposed to dynamically adjust the distribution of the reference vectors according to the scales of the objective functions. Our experimental results on a variety of benchmark test problems show that the proposed algorithm is highly competitive in comparison with five state-of-the-art EAs for many-objective optimization. In addition, we show that reference vectors are effective and cost-efficient for preference articulation, which is particularly desirable for many-objective optimization. Furthermore, a reference vector regeneration strategy is proposed for handling irregular PFs. Finally, the proposed algorithm is extended for solving constrained many-objective optimization problems.

On Test Functions for Evolutionary Multi-objective Optimization

In order to evaluate the relative performance of optimization algorithms benchmark problems are frequently used. In the case of multi-objective optimization (MOO), we will show in this paper that most known benchmark problems belong to a constrained class of functions with piecewise linear Pareto fronts in the parameter space. We present a straightforward way to define benchmark problems with an arbitrary Pareto front both in the fitness and parameter spaces. Furthermore, we introduce a difficulty measure based on the mapping of probability density functions from parameter to fitness space. Finally, we evaluate two MOO algorithms for new benchmark problems.

Managing approximate models in evolutionary aerodynamic design optimization

Approximate models have to be used in evolutionary optimization when the original fitness function is computationally very expensive. Unfortunately, the convergence property of the evolutionary algorithm is unclear when an approximate model is used for fitness evaluation because approximation errors are involved in the model. What is worse, the approximate model may introduce false optima that lead the evolutionary algorithm to a wrong solution. To address this problem, individual and generation based evolution control are introduced to ensure that the evolutionary algorithm using approximate fitness functions will converge correctly. A framework for managing approximate models in generation-based evolution control is proposed. This framework is well suited for parallel evolutionary optimization in which evaluation of the fitness function is time-consuming. Simulations on two benchmark problems and one example of aerodynamic design optimization demonstrate that the proposed algorithm is able to achieve a correct solution as well as a significantly reduced computation time.

Adaptive encoding for aerodynamic shape optimization using evolution strategies

The evaluation of fluid dynamic properties of various different structures is a computationally very demanding process. This is of particular importance when population based evolutionary algorithms are used for the optimization of aerodynamic structures like wings or turbine blades. Besides choosing algorithms which only need few generations or function evaluations, it is important to reduce the number of object parameters as much as possible. This is usually done by restricting the optimization to certain attributes of the design which are seen as important. By doing so, the freedom for the optimization is restricted to areas of the design space where good solutions are expected. This can be problematic especially if the properties of the design and their interactions are not known sufficiently well like for example for transonic flow conditions. In order to be able to combine the conflicting constraints of a minimal set of parameters and the maximal degree of freedom, we propose an adaptive or growing representation for spline coded structures. In this way, the optimization is started with a simple representation with a minimal description length. The number of describing parameters is adapted during the optimization using a mutation operator working on the structure of the encoding. We compare this method with four different evolution strategies using a spline fitting problem as a test function.

Voronoi-based estimation of distribution algorithm for multi-objective optimization

The distribution of the Pareto-optimal solutions often has a clear structure. To adapt evolutionary algorithms to the structure of a multi-objective optimization problem, either an adaptive representation or adaptive genetic operators should be employed. We suggest an estimation of distribution algorithm for solving multi-objective optimization, which is able to adjust its reproduction process to the problem structure. For this purpose, a new algorithm called Voronoi-based estimation of distribution algorithm (VEDA) is proposed. In VEDA, a Voronoi diagram is used to construct stochastic models, based on which new offspring will be generated. Empirical comparisons of the VEDA with other estimation of distribution algorithms (EDAs) and the popular NSGA-II algorithm are carried out. In addition, representation of Pareto-optimal solutions using a mathematical model rather than a solution set is also discussed.

Advanced High Turning Compressor Airfoils for Low Reynolds Number Condition: Part 1 — Design and Optimization

High performance compressor airfoils at a low Reynolds number condition at ~Re51.3310 5 ! have been developed using evolutionary algorithms in order to improve the performance of the outlet guide vane (OGV), used in a single low pressure turbine (LPT) of a small turbofan engine for business jet aircrafts. Two different numerical optimization methods, the evolution strategy (ES) and the multi-objective genetic algorithm (MOGA), were adopted for the design process to minimize the total pressure loss and the deviation angle at the design point at low Reynolds number condition. Especially, with respect to the MOGA, robustness against changes of the incidence angle is considered. The optimization process includes the representation of the blade geometry, the generation of a numerical grid and a blade-to-blade analysis using a quasi-three-dimensional Navier-Stokes solver with a k-v turbulence model including a newly implemented transition model to evaluate the performance. Overall aerodynamic performance and boundary layer properties for the two optimized blades are discussed numerically. The superior performance of the two optimized airfoils is demonstrated by a comparison with conventional controlled diffusion airfoils (CDA). The advantage in performance has been confirmed by detailed experimental investigations, which are presented in Part II of this paper. @DOI: 10.1115/1.1737780#

Advanced High Turning Compressor Airfoils for Low Reynolds Number Condition—Part I: Design and Optimization

High performance compressor airfoils at a low Reynolds number condition at (Re = 1.3 × 10 5 ) have been developed using evolutionary algorithms in order to improve the performance of the outlet guide vane (OGV), used in a single low pressure turbine (LPT) of a small turbofan engine for business jet aircrafts. Two different numerical optimization methods, the evolution strategy (ES) and the multi-objective genetic algorithm (MOGA), were adopted for the design process to minimize the total pressure loss and the deviation angle at the design point at low Reynolds number condition. Especially, with respect to the MOGA, robustness against changes of the incidence angle is considered. The optimization process includes the representation of the blade geometry, the generation of a numerical grid and a blade-to-blade analysis using a quasi-three-dimensional Navier-Stokes solver with a k-ω turbulence model including a newly implemented transition model to evaluate the performance. Overall aerodynamic performance and boundary layer properties for the two optimized blades are discussed numerically. The superior performance of the two optimized airfoils is demonstrated by a comparison with conventional controlled diffusion airfoils (CDA). The advantage in performance has been confirmed by detailed experimental investigations, which are presented in Part II of this paper.

Knowledge Extraction from Aerodynamic Design Data and its Application to 3D Turbine Blade Geometries

Applying numerical optimisation methods in the field of aerodynamic design optimisation normally leads to a huge amount of heterogeneous design data. While most often only the most promising results are investigated and used to drive further optimisations, general methods for investigating the entire design dataset are rare. We propose methods that allow the extraction of comprehensible knowledge from aerodynamic design data represented by discrete unstructured surface meshes. The knowledge is prepared in a way that is usable for guiding further computational as well as manual design and optimisation processes. A displacement measure is suggested in order to investigate local differences between designs. This measure provides information on the amount and direction of surface modifications. Using the displacement data in conjunction with statistical methods or data mining techniques provides meaningful knowledge from the dataset at hand. The theoretical concepts have been applied to a data set of 3D turbine stator blade geometries. The results have been verified by means of modifying the turbine blade geometry using direct manipulation of free form deformation (DMFFD) techniques. The performance of the deformed blade design has been calculated by running computational fluid dynamic (CFD) simulations. It is shown that the suggested framework provides reasonable results which can directly be transformed into design modifications in order to guide the design process.

Neural Networks for Fitness Approximation in Evolutionary Optimization

Approximate models such as neural networks are very helpful in evolutionary optimization when the original fitness function is computationally very expensive. This chapter presents a general introduction to methods for using approximate models in conjunction with the original fitness function. Individual and generation based evolution control is introduced to ensure that evolutionary algorithms using approximate fitness functions will converge to the true optimum. Frameworks for managing approximate models with generation-based or individualbased evolution control are described. To improve the approximation quality of the neural networks, techniques for optimizing the structure optimization of neural networks and for generating neural network ensembles are presented. The frameworks are illustrated on benchmark problems as well as on an example of aerodynamic design optimization.