Boris Naujoks

SMS-EMOA : Multiobjective selection based on dominated hypervolume

Abstract The hypervolume measure (or S metric) is a frequently applied quality measure for comparing the results of evolutionary multiobjective optimisation algorithms (EMOA). The new idea is to aim explicitly for the maximisation of the dominated hypervolume within the optimisation process. A steady-state EMOA is proposed that features a selection operator based on the hypervolume measure combined with the concept of non-dominated sorting. The algorithm’s population evolves to a well-distributed set of solutions, thereby focussing on interesting regions of the Pareto front. The performance of the devised S metric selection EMOA (SMS-EMOA) is compared to state-of-the-art methods on two- and three-objective benchmark suites as well as on aeronautical real-world applications.

An EMO algorithm using the hypervolume measure as selection criterion

The hypervolume measure is one of the most frequently applied measures for comparing the results of evolutionary multiobjective optimization algorithms (EMOA). The idea to use this measure for selection is self-evident. A steady-state EMOA will be devised, that combines concepts of non-dominated sorting with a selection operator based on the hypervolume measure. The algorithm computes a well distributed set of solutions with bounded size thereby focussing on interesting regions of the Pareto front(s). By means of standard benchmark problems the algorithm will be compared to other well established EMOA. The results show that our new algorithm achieves good convergence to the Pareto front and outperforms standard methods in the hypervolume covered. We also studied the applicability of the new approach in the important field of design optimization. In order to reduce the number of time consuming precise function evaluations, the algorithm will be supported by approximate function evaluations based on Kriging metamodels. First results on an airfoil redesign problem indicate a good performance of this approach, especially if the computation of a small, bounded number of well-distributed solutions is desired.

Pareto-, aggregation-, and indicator-based methods in many-objective optimization

Research within the area of Evolutionary Multi-objective Optimization (EMO) focused on two- and three-dimensional objective functions, so far. Most algorithms have been developed for and tested on this limited application area. To broaden the insight in the behavior of EMO algorithms (EMOA) in higher dimensional objective spaces, a comprehensive benchmarking is presented, featuring several state-of-the-art EMOA, as well as an aggregative approach and a restart strategy on established scalable test problems with three to six objectives. It is demonstrated why the performance of well-established EMOA (NSGAII, SPEA2) rapidly degradates with increasing dimension. Newer EMOA like Ɛ -MOEA, MSOPS, IBEA and SMS-EMOA cope very well with high-dimensional objective spaces. Their specific advantages and drawbacks are illustrated, thus giving valuable hints for practitioners which EMOA to choose depending on the optimization scenario. Additionally, a new method for the generation of weight vectors usable in aggregation methods is presented.

Single- and multiobjective evolutionary optimization assisted by Gaussian random field metamodels

This paper presents and analyzes in detail an efficient search method based on evolutionary algorithms (EA) assisted by local Gaussian random field metamodels (GRFM). It is created for the use in optimization problems with one (or many) computationally expensive evaluation function(s). The role of GRFM is to predict objective function values for new candidate solutions by exploiting information recorded during previous evaluations. Moreover, GRFM are able to provide estimates of the confidence of their predictions. Predictions and their confidence intervals predicted by GRFM are used by the metamodel assisted EA. It selects the promising members in each generation and carries out exact, costly evaluations only for them. The extensive use of the uncertainty information of predictions for screening the candidate solutions makes it possible to significantly reduce the computational cost of singleand multiobjective EA. This is adequately demonstrated in this paper by means of mathematical test cases and a multipoint airfoil design in aerodynamics

Capabilities of EMOA to detect and preserve equivalent pareto subsets

Recent works in evolutionary multiobjective optimization suggest to shift the focus from solely evaluating optimization success in the objective space to also taking the decision space into account. They indicate that this may be a) necessary to express the users requirements of obtaining distinct solutions (distinct Pareto set parts or subsets) of similar quality (comparable locations on the Pareto front) in real-world applications, and b) a demanding task for the currently most commonly used algorithms.We investigate if standard EMOA are able to detect and preserve equivalent Pareto subsets and develop an own special purpose EMOA that meets these requirements reliably.

Metamodel Assisted Multiobjective Optimisation Strategies and their Application in Airfoil Design

In this paper various metamodel-assisted multiobjective evolutionary algorithms (M-MOEA) for optimisation with time-consuming function evaluations are proposed and studied. Gaussian field (Kriging) models fitted by results from previous evaluations are used in order to pre-screen candidate solutions and decide whether they should be rejected or evaluated precisely. The approximations provide upper and lower bound estimations for the true function values. Three different rejection principles are proposed, discussed and integrated into recent MOEA variants (NSGA-II and ∈-MOEA). Experimental studies on a theoretical test case and in airfoil design demonstrate the improvements in diversity of solutions and convergence to the pareto fronts that can be achieved by using metamodels for pre-screening.

Enhancing Decision Space Diversity in Evolutionary Multiobjective Algorithms

In multi-criterion optimization, Pareto-optimal solutions that appear very similar in the objective space may have very different pre-images. In many practical applications the decision makers, who select a solution or preferred region on the Pareto-front, may want to know different pre-images of the selected solutions. Especially, this will be the case when they would like to present alternative design candidates in later stages of a multidisciplinary design process. n nIn this paper we extend an existing CMA-ES niching framework, which has been previously applied successfully to multi-modal optimization, to the multi-criterion domain for boosting decision space diversity. At the same time, we introduce the concept of space aggregation for diversity maintenance in the aggregated spaces, i.e. search/decision and objective space. Empirical results on synthetic multi-modal bi-criteria test problems with known efficient sets and Pareto-fronts demonstrate that the diversity in the decision space can be significantly enhanced without hampering the convergence to a precise and diverse Pareto front approximation in the objective space of the original algorithm.

Pareto set and EMOA behavior for simple multimodal multiobjective functions

Recent research on evolutionary multiobjective optimization has mainly focused on Pareto fronts. However, we state that proper behavior of the utilized algorithms in decision/search space is necessary for obtaining good results if multimodal objective functions are concerned. Therefore, it makes sense to observe the development of Pareto sets as well. We do so on a simple, configurable problem, and detect interesting interactions between induced changes to the Pareto set and the ability of three optimization algorithms to keep track of Pareto fronts.

Towards Intelligent Team Composition and Maneuvering in Real-Time Strategy Games

Players of real-time strategy (RTS) games are often annoyed by the inability of the game AI to select and move teams of units in a natural way. Units travel and battle separately, resulting in huge losses and the AI looking unintelligent, as can the choice of units sent to counteract the opponents. Players are affected as well as computer commanded factions because they cannot micromanage all team related issues. We suggest improving AI behavior by combining well-known computational intelligence techniques applied in an original way. Team composition for battling spatially distributed opponent groups is supported by a learning self-organizing map (SOM) that relies on an evolutionary algorithm (EA) to adapt it to the game. Different abilities of unit types are thus employed in a near-optimal way, reminiscent of human ad hoc decisions. Team movement is greatly enhanced by flocking and influence map-based path finding, leading to a more natural behavior by preserving individual motion types. The team decision to either attack or avoid a group of enemy units is easily parametrizable, incorporating team characteristics from fearful to daredevil. We demonstrate that these two approaches work well separately, but also that they go together naturally, thereby leading to an improved and flexible group behavior.

OCD: Online Convergence Detection for Evolutionary Multi-Objective Algorithms Based on Statistical Testing

Over the last decades, evolutionary algorithms (EA) have proven their applicability to hard and complex industrial optimization problems in many cases. However, especially in cases with high computational demands for fitness evaluations (FE), the number of required FE is often seen as a drawback of these techniques. This is partly due to lacking robust and reliable methods to determine convergence, which would stop the algorithm before useless evaluations are carried out. To overcome this drawback, we define a method for online convergence detection (OCD) based on statistical tests, which invokes a number of performance indicators and which can be applied on a stand-alone basis (no predefined Pareto fronts, ideal and reference points). Our experiments show the general applicability of OCD by analyzing its performance for different algorithmic setups and on different classes of test functions. Furthermore, we show that the number of FE can be reduced considerably --- compared to common suggestions from literature --- without significantly deteriorating approximation accuracy.