New Uncertainty Handling Strategies in Multi-objective Evolutionary Optimization - Voß, Thomas; Trautmann, Heike; Igel, Christian (Parallel Problem Solving from Nature, PPSN XI)
Since many real-world optimization problems are noisy, vector optimization algorithms that can cope with noise and uncertainty are required. We propose new, robust selection strategies for evolutionary multi-objective optimization in the presence of noise. We apply new measures of uncertainty for estimating the recently introduced Pareto-dominance for uncertain and noisy environments (PDU). The first measure is the inter-quartile range of the outcomes of repeated function evaluations. The second is based on axis-aligned bounding boxes around the upper and lower quantiles of the sampled fitness values in objective space. Experiments on real and artificial problems show promising results.

Integration of Preferences in Hypervolume-Based Multiobjective Evolutionary Algorithms by Means of Desirability Functions - Wagner, Tobias; Trautmann, Heike (IEEE Transactions on Evolutionary Computation)
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.

Work roll cooling system design optimisation in presence of uncertainty and constrains - Azene, Y.T.; Roy, R.; Farrugia, D.; Onisa, C.; Mehnen, J.; Trautmann, Heike (CIRP Journal of Manufacturing Science and Technology)
The paper presents a framework to optimise the design of work roll based on the cooling performance. The framework develops meta-models from a set of finite element analyses (FEA) of the roll cooling. A design of experiment technique is used to identify the FEA runs. The research also identifies sources of uncertainties in the design process. A robust evolutionary multi-objective evaluation technique is applied to the design optimisation in constrained problems with real life uncertainty. The approach handles uncertainties associated both with design variables and fitness functions. Constraints violation within the neighbourhood of a design is considered as part of a measurement for degree of feasibility and robustness of a solution.
Keywords: Roll cooling design; Uncertainty; Design optimisation; Multi-objective optimisation; Constraint in design

Online convergence detection for evolutionary multi-objective algorithms revisited - Wagner, Tobias; Trautmann, Heike (IEEE Congress on Evolutionary Computation)
The design and application of termination criteria has become an important aspect in evolutionary multi-objective optimization. Online convergence detection (OCD) determines when further generations are no longer promising based on statistical tests on a set of performance indicators. The behavior of OCD mainly depends on two parameters, the number of preceding generations considered in the statistical tests and the desired variance limit. In this paper, guidelines for selecting appropriate combinations of these parameters are empirically derived based on design-of-experiment methods. Furthermore, a variant of OCD is introduced which directly operates on the hypervolume indicator - the internal measure of the SMS-EMOA. This allows a separated analysis of the variance criterion and reduces the complexity of OCD. Based on the experimental design, a systematic comparison with the classical OCD approach is performed and differences between the appropriate parameterizations of both variants are highlighted.

OCD: Online Convergence Detection for Evolutionary Multi-Objective Algorithms Based on Statistical Testing - Wagner, Tobias; Trautmann, Heike; Naujoks, Boris (Evolutionary Multi-Criterion Optimization)
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.

Pareto-dominance in noisy environments - Trautmann, Heike; Mehnen, Jorn; Naujoks, Boris (2009 IEEE Congress on Evolutionary Computation)
Noisy environments are a challenging task for multiobjective evolutionary algorithms. The algorithms may be trapped in local optima or even become a random search in the decision and objective space. In the course of the paper the classical definition of Pareto-dominance is enhanced subject to noisy objective functions in order to make the evolutionary search process more robust and to generate a reliable Pareto front. At each point in the decision space the objective functions are evaluated a fixed number of times and the convex hull of the objective function vectors is computed. Expectation is associated with the median of the objective function values while uncertainty is reflected by the average distance of the median in each dimension to the points defining the convex hull. By combining these two indicators a new concept of Pareto-dominance is set up. An implementation in NSGA-II and application to test problems show a gain in robustness and search quality.

Online convergence detection for multiobjective aerodynamic applications - Naujoks, Boris; Trautmann, Heike (2009 IEEE Congress on Evolutionary Computation)
Industry applications of multiobjective optimization problems mostly are characterized by the demand for high quality solutions on the one hand. On the other hand an optimization result is desired which at any rate meets the time constraints for the evolutionary multiobjective algorithms (EMOA). The handling of this trade-off is a frequently discussed issue in multiobjective evolutionary optimization.

Statistical Methods for Improving Multi-objective Evolutionary Optimisation - Trautmann, Heike; Mehnen, J (International Journal of Computational Intelligence Research)

Preference-based Pareto optimization in certain and noisy environments - Trautmann, Heike; Mehnen, Jörn (Engineering Optimization)
In this article a method for including a priori preferences of decision makers into multicriteria optimization problems is presented. A set of Pareto-optimal solutions is determined via desirability functions of the objectives which reveal experts’ preferences regarding different objective regions. An application to noisy objective functions is not straightforward but very relevant for practical applications. Two approaches are introduced in order to handle the respective uncertainties by means of the proposed preference-based Pareto optimization. By applying the methods to the original and uncertain Binh problem and a noisy single cut turning cost optimization problem, these approaches prove to be very effective in focusing on different parts of the Pareto front of the ori-ginal problem in both certain and noisy environments.

Statistical Methods for Convergence Detection of Multi-Objective Evolutionary Algorithms - Trautmann, Heike; Wagner, T.; Naujoks, B.; Preuss, M.; Mehnen, J. (Evolutionary Computation)
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.

Robust Multi-objective Optimisation of Weld Bead Geometry for Additive Manufacturing - Mehnen, J; Trautmann, Heike (Proceedings of the 6$^th$ CIRP International Seminar on Intelligent Computation in Manufacturing Engineering (CIRP ICME ’08))

A Convergence Criterion for Multiobjective Evolutionary Algorithms Based on Systematic Statistical Testing - Trautmann, Heike; Ligges, Uwe; Mehnen, Jörn; Preuss, Mike (Parallel Problem Solving from Nature – PPSN X)
A systematic approach for determining the generation number at which a specific Multi-Objective Evolutionary Algorithm (MOEA) has converged for a given optimization problem is introduced. Convergence is measured by the performance indicators Generational Distance, Spread and Hypervolume. The stochastic nature of the MOEA is taken into account by repeated runs per generation number which results in a highly robust procedure. For each generation number the MOEA is repeated a fixed number of times, and the Kolmogorow-Smirnov-Test is used in order to decide if a significant change in performance is gained in comparison to preceding generations. A comparison of different MOEAs on a problem with respect to necessary generation numbers becomes possible, and the understanding of the algorithm’s behaviour is supported by analysing the development of the indicator values. The procedure is illustrated by means of standard test problems.

Introducing user preference using Desirability Functions in Multi-Objective Evolutionary Optimisation of noisy processes - Mehnen, Jorn; Trautmann, Heike; Tiwari, Ashutosh (2007 IEEE Congress on Evolutionary Computation)
Multi-objective evolutionary algorithms (MOEAs) are generally designed to find a well spread Pareto-front approximation. Often, only a small section of this front may be of practical interest. Desirability functions (DFs) are able to describe user preferences intuitively. Furthermore, DFs can be attached to any fitness function easily. This way, desirability functions can help in guiding MOEAs without introducing additional restrictions or changes to the algorithm. The application of noisy fitness functions is not straight forward but relevant to many real-world problems. Therefore, a variant of Harrington’s one-sided desirability function using expectations is introduced which takes noise into account. A deterministic strategy as well as the XSGA-II are used in combination with DF to solve a noisy Binh problem and a noisy cost estimation problem for turning processes.

On the Distribution of the Desirability Index using Harrington’s Desirability Function - Trautmann, Heike; Weihs, C (Metrika)
The concept of desirability is a means for complexity reduction of multivariate quality optimization. This paper provides a theoretical breakthrough regarding desirability indices, which application fields were formerly limited primarily by the lack of its distribution. Focussed are the distributions of Harrington’s desirability functions and different types of the desirability index.

Integration of Expert’s Preferences in Pareto Optimization by Desirability Function Techniques - Mehnen, J; Trautmann, Heike (CIRP ICME ’06) — Proceedings of the 5$^th$ CIRP International Seminar on Intelligent Computation in Manufacturing Engineering)