Tobias Wagner

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.

On the properties of the R2 indicator

In multiobjective optimization, set-based performance indicators are commonly used to assess the quality of a Pareto front approximation. Based on the scalarization obtained by these indicators, a performance comparison of multiobjective optimization algorithms becomes possible. The R2 and the Hypervolume (HV) indicator represent two recommended approaches which have shown a correlated behavior in recent empirical studies. Whereas the HV indicator has been comprehensively analyzed in the last years, almost no studies on the R2 indicator exist. In this paper, we thus perform a comprehensive investigation of the properties of the R2 indicator in a theoretical and empirical way. The influence of the number and distribution of the weight vectors on the optimal distribution of μ solutions is analyzed. Based on a comparative analysis, specific characteristics and differences of the R2 and HV indicator are presented.

Integration of Preferences in Hypervolume-Based Multiobjective Evolutionary Algorithms by Means of Desirability Functions

In this paper, a concept for efficiently approximating the practically relevant regions of the Pareto front (PF) is introduced. Instead of the original objectives, desirability functions (DFs) of the objectives are optimized, which express the preferences of the decision maker. The original problem formulation and the optimization algorithm do not have to be modified. DFs map an objective to the domain [0, 1] and nonlinearly increase with better objective quality. By means of this mapping, values of different objectives and units become comparable. A biased distribution of the solutions in the PF approximation based on different scalings of the objectives is prevented. Thus, we propose the integration of DFs into the S-metric selection evolutionary multiobjective algorithm. The transformation ensures the meaning of the hypervolumes internally computed. Furthermore, it is shown that the reference point for the hypervolume calculation can be set intuitively. The approach is analyzed using standard test problems. Moreover, a practical validation by means of the optimization of a turning process is performed.

On expected-improvement criteria for model-based multi-objective optimization

Surrogate models, as used for the Design and Analysis of Computer Experiments (DACE), can significantly reduce the resources necessary in cases of expensive evaluations. They provide a prediction of the objective and of the corresponding uncertainty, which can then be combined to a figure of merit for a sequential optimization. In singleobjective optimization, the expected improvement (EI) has proven to provide a combination that balances successfully between local and global search. Thus, it has recently been adapted to evolutionary multi-objective optimization (EMO) in different ways. In this paper, we provide an overview of the existing EI extensions for EMO and propose new formulations of the EI based on the hypervolume. We set up a list of necessary and desirable properties, which is used to reveal the strengths and weaknesses of the criteria by both theoretical and experimental analyses.

Clustered multiple generalized expected improvement: A novel infill sampling criterion for surrogate models

Surrogate model-based optimization is a well-known technique for optimizing expensive black-box functions. By applying this function approximation, the number of real problem evaluations can be reduced because the optimization is performed on the model. In this case two contradictory targets have to be achieved: increasing global model accuracy and exploiting potentially optimal areas. The key to these targets is the criterion for selecting the next point, which is then evaluated on the expensive black-box function - the dasiainfill sampling criterionpsila. Therefore, a novel approach - the dasiaClustered Multiple Generalized Expected Improvementpsila (CMGEI) - is introduced and motivated by an empirical study. Furthermore, experiments benchmarking its performance compared to the state of the art are presented.

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.

Statistical methods for convergence detection of multi-objective evolutionary algorithms

In this paper, two approaches for estimating the generation in which a multi-objective evolutionary algorithm (MOEA) shows statistically significant signs of convergence are introduced. A set-based perspective is taken where convergence is measured by performance indicators. The proposed techniques fulfill the requirements of proper statistical assessment on the one hand and efficient optimisation for real-world problems on the other hand. The first approach accounts for the stochastic nature of the MOEA by repeating the optimisation runs for increasing generation numbers and analysing the performance indicators using statistical tools. This technique results in a very robust offline procedure. Moreover, an online convergence detection method is introduced as well. This method automatically stops the MOEA when either the variance of the performance indicators falls below a specified threshold or a stagnation of their overall trend is detected. Both methods are analysed and compared for two MOEA and on different classes of benchmark functions. It is shown that the methods successfully operate on all stated problems needing less function evaluations while preserving good approximation quality at the same time.

A taxonomy of online stopping criteria for multi-objective evolutionary algorithms

The use of multi-objective evolutionary algorithms for solving black-box problems with multiple conflicting objectives has become an important research area. However, when no gradient information is available, the examination of formal convergence or optimality criteria is often impossible. Thus, sophisticated heuristic online stopping criteria (OSC) have recently become subject of intensive research. In order to establish formal guidelines for a systematic research, we present a taxonomy of OSC in this paper.We integrate the known approaches within the taxonomy and discuss them by extracting their building blocks. The formal structure of the taxonomy is used as a basis for the implementation of a comprehensive MATLAB toolbox. Both contributions, the formal taxonomy and the MATLAB implementation, provide a framework for the analysis and evaluation of existing and new OSC approaches.

2 indicator-based multiobjective search

In multiobjective optimization, set-based performance indicators are commonly used to assess the quality of a Pareto front approximation. Based on the scalarization obtained by these indicators, a performance comparison of multiobjective optimization algorithms becomes possible. The and the hypervolume (HV) indicator represent two recommended approaches which have shown a correlated behavior in recent empirical studies. Whereas the HV indicator has been comprehensively analyzed in the last years, almost no studies on the indicator exist. In this extended version of our previous conference paper, we thus perform a comprehensive investigation of the properties of the indicator in a theoretical and empirical way. The influence of the number and distribution of the weight vectors on the optimal distribution of solutions is analyzed. Based on a comparative analysis, specific characteristics and differences of the and HV indicator are presented. Furthermore, the indicator is integrated into an indicator-based steady-state evolutionary multiobjective optimization algorithm (EMOA). It is shown that the so-called -EMOA can accurately approximate the optimal distribution of solutions regarding .

Empirical modeling of hard turning of AISI 6150 steel using design and analysis of computer experiments

In the present paper an experimental study to investigate the turning of hardened AISI 6150 heat treatable steel using polycrystalline boron nitride (PCBN) tools is presented. Design and analysis of computer experiments (DACE) was used to generate a comprehensive empirical description of the process characteristics. More specific, the effects of the parameters cutting speed, feed and depth of cut on the objectives tool wear, tool life, tool life volume, surface finish and process forces were modeled. A total of 157 experiments was carried out with 15 different parameter-value sets to obtain the training data for modeling the progression of the objectives versus cutting path length and width of flank wear land. Pseudo-3D surface plots are generated to visualize the effects and interactions. Unexpected effects of depth of cut on tool life were found and the validity of conclusions about the effect of cutting speed on tool wear and tool life are discussed. Moreover, qualitative explanations for some of the observed effects are presented.