R2 Indicator Based Multiobjective Search - Brockhoff, D; Wagner, T; Trautmann, Heike (Evolutionary Computation Journal)
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 extended version of our previous conference 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. Furthermore, the R2 indicator is integrated into an indicator-based steady-state evolutionary multiobjective optimization algorithm (EMOA). It is shown that the so-called R2-EMOA can accurately approximate the optimal distribution of µ solutions regarding R2.

Multi-Objective Analysis of Approaches to Dynamic Routing Of a Vehicle - Grimme, C; Meisel, S; Trautmann, Heike; Rudolph, G; Wölck, M (Proceedings of the European Conference On Information Systems)
We consider a routing problem for a single vehicle serving customer Locations in the course of time. A subset of these customers must necessarily be served, while the complement of this subset contains dynamic customers which request for service over time, and which do not necessarily need to be served. The decision maker’s conflicting goals are serving as many customers as possible as well as minimizing total travel distance. We solve this bi-objective Problem with an evolutionary multi-objective algorithm in order to provide an a-posteriori evaluation tool for enabling decision makers to assess the single objective solution strategies that they actually use in real-time. We present the modifications to be applied to the evolutionary multi-objective algorithm NSGA2 in order to solve the routing problem, we describe a number of real-time single-objective solution strategies, and we finally use the gained efficient trade-off solutions of NSGA2 to exemplarily evaluate the real-time strategies. Our results show that the evolutionary multi-objective approach is well-suited to generate benchmarks for assessing dynamic heuristic strategies. Our findings point into future directions for designing dynamic multi-objective approaches for the vehicle routing problem with time windows.

Analyzing the BBOB Results by Means of Benchmarking Concepts - Mersmann, O; Preuss, M; Trautmann, Heike; Bischl, B; Weihs, C (Evolutionary Computation Journal)
We present methods to answer two basic questions that arise when benchmarking optimization algorithms. The first one is: which algorithm is the "best" one? and the second one is: which algorithm should I use for my real-world problem? Both are connected and neither is easy to answer. We present a theoretical framework for designing and analyzing the raw data of such benchmark experiments. This represents a first step in answering the aforementioned questions. The 2009 and 2010 BBOB benchmark results are analyzed by means of this framework and we derive insight regarding the answers to the two questions. Furthermore, we discuss how to properly aggregate rankings from algorithm evaluations on individual problems into a consensus, its theoretical background and which common pitfalls should be avoided. Finally, we address the grouping of test problems into sets with similar optimizer rankings and investigate whether these are reflected by already proposed test problem characteristics, finding that this is not always the case.

Evaluation of a Multi-Objective EA on Benchmark Instances for Dynamic Routing of a Vehicle - Meisel, Stephan; Grimme, Christian; Bossek, Jakob; Wölck, Martin; Rudolph, Guenter; Trautmann, Heike (Proceedings of the Genetic and Evolutionary Computation Conference (GECCO ’15))
We evaluate the performance of a multi-objective evolutionary algorithm on a class of dynamic routing problems with a single vehicle. In particular we focus on relating algorithmic performance to the most prominent characteristics of problem instances. The routing problem considers two types of customers: mandatory customers must be visited whereas optional customers do not necessarily have to be visited. Moreover, mandatory customers are known prior to the start of the tour whereas optional customers request for service at later points in time with the vehicle already being on its way. The multi-objective optimization problem then results as maximizing the number of visited customers while simultaneously minimizing total travel time. As an a-posteriori evaluation tool, the evolutionary algorithm aims at approximating the related Pareto set for specifically designed benchmarking instances differing in terms of number of customers, geographical layout, fraction of mandatory customers, and request times of optional customers. Conceptional and experimental comparisons to online heuristic procedures are provided.

Cell Mapping Techniques for Exploratory Landscape Analysis - Kerschke, Pascal; Preuss, Mike; Hernández, Carlos; Schütze, Oliver; Sun, Jian-Qiao; Grimme, Christian; Rudolph, Günter; Bischl, Bernd; Trautmann, Heike (EVOLVE — A Bridge between Probability, Set Oriented Numerics, and Evolutionary Computation V)
Exploratory Landscape Analysis is an effective and sophisticated approach to characterize the properties of continuous optimization problems. The overall aim is to exploit this knowledge to give recommendations of the individually best suited algorithm for unseen optimization problems. Recent research revealed a high potential of this methodology in this respect based on a set of well-defined, computable features which only requires a quite small sample of function evaluations. In this paper, new features based on the cell mapping concept are introduced and shown to improve the existing feature set in terms of predicting expert-designed high-level properties, such as the degree of multimodality or the global structure, for 2-dimensional single objective optimization problems.

A Multiobjective Evolutionary Algorithm Guided by Averaged Hausdorff Distance to Aspiration Sets - Rudolph, G; Schütze, O; Grimme, C; Trautmann, Heike (EVOLVE — A Bridge between Probability, Set Oriented Numerics, and Evolutionary Computation V)
The incorporation of expert knowledge into multiobjective optimization is an important issue which in this paper is reflected in terms of an aspiration set consisting of multiple reference points. The behaviour of the recently introduced evolutionary multiobjective algorithm AS-EMOA is analysed in detail and comparatively studied for bi-objective optimization problems w.r.t. R-NSGA2 and a respective variant. It will be shown that the averaged Hausdorff distance, integrated into AS-EMOA, is an efficient means to accurately approximate the desired aspiration set.

An Aspiration Set EMOA Based on Averaged Hausdorff Distances - Rudolph, Günter; Grimme, Christian; Schütze, Oliver; Trautmann, Heike (Proceedings of the Learning and Intelligent OptimizatioN Conference (LION 8))
We propose an evolutionary multiobjective algorithm that approximates multiple reference points (the aspiration set) in a single run using the concept of the averaged Hausdorff distance.

Stopping Criteria for Multimodal Optimization - Wessing, S; Preuss, M; Trautmann, Heike (Proceedings of the Parallel Problem Solving from Nature — PPSN XIII)
Multimodal optimization requires maintenance of a good search space coverage and approximation of several optima at the same time. We analyze two constitutive optimization algorithms and show that in many cases, a phase transition occurs at some point, so that either diversity collapses or optimization stagnates. But how to derive suitable stopping criteria for multimodal optimization? Experimental results indicate that an algorithm’s population contains sufficient information to estimate the point in time when several performance indicators reach their optimum. Thus, stopping criteria are formulated based on summary characteristics employing objective values and mutation strength.

The Directed Search Method for Pareto Front Approximations with Maximum Dominated Hypervolume - Sosa, Hernández V; Schütze, O; Rudolph, G; Trautmann, Heike (EVOLVE — A Bridge between Probability, Set Oriented Numerics, and Evolutionary Computation IV)
In many applications one is faced with the problem that multiple objectives have to be optimized at the same time. Since typically the solution set of such multi-objective optimization problems forms a manifold which cannot be computed analytically, one is in many cases interested in a suitable finite size approximation of this set. One widely used approach is to find a representative set that maximizes the dominated hypervolume that is defined by the images in objective space of these solutions and a given reference point. In this paper, we propose a new point-wise iterative search procedure, Hypervolume Directed Search (HVDS), that aims to increase the hypervolume of a given point in an archive for bi-objective unconstrained optimization problems. We present the HVDS both as a standalone algorithm and as a local searcher within a specialized evolutionary algorithm. Numerical results confirm the strength of the novel approach.

Finding Evenly Spaced Pareto Fronts for Three-Objective Optimization Problems - Trautmann, Heike; Rudolph, G; Dominguez-Medina, C; Schütze, O (EVOLVE — A Bridge between Probability, Set Oriented Numerics, and Evolutionary Computation II)
The averaged Hausdorff distance Δ p is a performance indicator in multi-objective evolutionary optimization which simultaneously takes into account proximity to the true Pareto front and uniform spread of solutions. Recently, the multi-objective evolutionary algorithm Δ p -EMOA was introduced which successfully generates evenly spaced Pareto front approximations for bi-objective problems by integrating an external archiving strategy into the SMS-EMOA based on Δ p . In this work a conceptual generalization of the Δ p -EMOA for higher objective space dimensions is presented and experimentally compared to state-of-the art EMOA as well as specialized EMOA variants on three-dimensional optimization problems.

A Feature-Based Comparison of Local Search and the Christofides Algorithm for the Travelling Salesperson Problem - Nallaperuma, Samadhi; Wagner, Markus; Neumann, Frank; Bischl, Bernd; Mersmann, Olaf; Trautmann, Heike (Proceedings of the Twelfth Workshop on Foundations of Genetic Algorithms XII)
Understanding the behaviour of well-known algorithms for classical NP-hard optimisation problems is still a difficult task. With this paper, we contribute to this research direction and carry out a feature based comparison of local search and the well-known Christofides approximation algorithm for the Traveling Salesperson Problem. We use an evolutionary algorithm approach to construct easy and hard instances for the Christofides algorithm, where we measure hardness in terms of approximation ratio. Our results point out important features and lead to hard and easy instances for this famous algorithm. Furthermore, our cross-comparison gives new insights on the complementary benefits of the different approaches.
Keywords: approximation algorithms; local search; traveling salesperson problem; feature selection; prediction; classification

Evenly Spaced Pareto Front Approximations for Tricriteria Problems Based on Triangulation - Rudolph, G; Trautmann, Heike; Sengupta, S; Schütze, O (Evolutionary Multi-Criterion Optimization — 7$^th$ International Conference, EMO 2013, Sheffield, UK, Proceedings)
In some technical applications like multiobjective online control an evenly spaced approximation of the Pareto front is desired. Since standard evolutionary multiobjective optimization (EMO) algorithms have not been designed for that kind of approximation we propose an archive-based plug-in method that builds an evenly spaced approximation using averaged Hausdorff measure between archive and reference front. In case of three objectives this reference font is constructed from a triangulated approximation of the Pareto front from a previous experiment. The plug-in can be deployed in online or offline mode for any kind of EMO algorithm.

Directed Search Method for Indicator-based Multi-objective Evolutionary Algorithms - Sosa-Hernandez, VA; Schütze, O; Rudoph, G; Trautmann, Heike (Proceeding of the Fifteenth Annual Conference Companion on Genetic and Evolutionary Computation Conference Companion)
Indicator based evolutionary algorithms have caught the interest of many researchers for the treatment of multi-objective optimization problems in the recent past since they deliver the desired approximation of the solution set and due to a usually better performance compared to dominance based algorithms. Nevertheless, these methods still suffer the drawback that many function evaluations are required to obtain a suitable representation of the solution set. The aim of this study is to present the Directed Search (DS) Method as local searcher within global indicator based optimization algorithms. For this, we will present the DS in the context of hypervolume maximization leading to both a new local search algorithm and a new memetic algorithm. Further, we will present first attempts to adapt the DS to a class of parameter dependent problems.

Evenly spaced Pareto fronts of quad-objective problems using PSA partitioning technique - Dominguez-Medina, C; Rudolph, G; Schütze, O; Trautmann, Heike (Proceedings of the 2013 IEEE Congress on Evolutionary Computation (CEC))
Here we address the problem of computing finite size Hausdorff approximations of the Pareto front of four-objective optimization problems by means of evolutionary computing. Since many applications desire an approximation evenly spread along the Pareto front and approximations that are good in the Hausdorff sense are typically evenly spread along the Pareto front we consider three different evolutionary multi-objective algorithms tailored to that purpose, where two of them are based on the Part and Selection Algorithm (PSA). Finally, we present some numerical results indicating the strength of the novel methods.

Reactive strategy choice in StarCraft by means of Fuzzy Control - Preuss, Mike; Kozakowski, Daniel; Hagelbäck, Johan; Trautmann, Heike (2013 IEEE Conference on Computational Inteligence in Games (CIG))
Current StarCraft bots are not very flexible in their strategy choice, most of them just follow a manually optimized one, usually a rush. We suggest a method of augmenting existing bots via Fuzzy Control in order to make them react on the current game situation. According to the available information, the best matching of a pool of strategies is chosen. While the method is very general and can be applied easily to many bots, we implement it for the existing BTHAI bot and show experimentally how the modifications affects its gameplay, and how it is improved compared to the original version.

Indicator-based Selection in Evolutionary Multiobjective Optimization Algorithms Based On the Desirability Index - Trautmann, Heike; Wagner, T; Biermann, D; Weihs, C (Journal of Multi-Criteria Decision Analysis)
In multiobjective optimization, the identification of practically relevant solutions on the Pareto-optimal front is an important research topic. Desirability functions (DFs) allow the preferences of the decision maker to be specified in an intuitive way. Recently, it has been shown for continuous optimization problems that an a priori transformation of the objectives by means of DFs can be used to focus the search of a hypervolume-based evolutionary algorithm on the desired part of the front. In many-objective optimization, however, the computational complexity of the hypervolume can become a crucial part. Thus, an alternative to this approach will be presented in this paper. The new algorithm operates in the untransformed objective space, but the desirability index (DI), that is, a DF-based scalarization, will be used as the second-level selection criterion in the non-dominated sorting. The diversity and uniform distribution of the resulting approximation are ensured by the use of an external archive. In the experiments, different preferences are specified as DFs, and their effects are investigated. It is shown that trade-off solutions are generated in the desired regions of the Pareto-optimal front and with a density adaptive to the DI. The efficiency of the approach with respect to increasing objective space dimension is also analysed using scalable test functions. The convergence speed is superior to other set-based and preference-based evolutionary multiobjective algorithms while the approach is of low computational complexity due to cheap DI evaluations. Copyright © 2013 John Wiley & Sons, Ltd.

Preference Articulation by Means of the R2 Indicator - Wagner, Tobias; Trautmann, Heike; Brockhoff, Dimo (Evolutionary Multi-Criterion Optimization)
In multi-objective optimization, set-based performance indicators have become the state of the art for assessing the quality of Pareto front approximations. As a consequence, they are also more and more used within the design of multi-objective optimization algorithms. The R2 and the Hypervolume (HV) indicator represent two popular examples. In order to understand the behavior and the approximations preferred by these indicators and algorithms, a comprehensive knowledge of the indicator’s properties is required. Whereas this knowledge is available for the HV, we presented a first approach in this direction for the R2 indicator just recently. In this paper, we build upon this knowledge and enhance the considerations with respect to the integration of preferences into the R2 indicator. More specifically, we analyze the effect of the reference point, the domain of the weights, and the distribution of weight vectors on the optimization of $μ$ solutions with respect to the R2 indicator. By means of theoretical findings and empirical evidence, we show the potentials of these three possibilities using the optimal distribution of $μ$ solutions for exemplary setups.

R2-EMOA: Focused Multiobjective Search Using R2-Indicator-Based Selection - Trautmann, Heike; Wagner, Tobias; Brockhoff, Dimo (Learning and Intelligent Optimization)
An indicator-based evolutionary multiobjective optimization algorithm (EMOA) is introduced which incorporates the contribution to the unary R2-indicator as the secondary selection criterion. First experiments indicate that the R2-EMOA accurately approximates the Pareto front of the considered continuous multiobjective optimization problems. Furthermore, decision makers’ preferences can be included by adjusting the weight vector distributions of the indicator which results in a focused search behavior.

A Novel Feature-Based Approach to Characterize Algorithm Performance for the Traveling Salesman Problem - Mersmann, O; Bischl, B; Trautmann, Heike; Wagner, M; Bossek, Jakob; Neumann, F (Annals of Mathematics and Artificial Intelligence)
Meta-heuristics are frequently used to tackle NP-hard combinatorial optimization problems. With this paper we contribute to the understanding of the success of 2-opt based local search algorithms for solving the traveling salesperson problem (TSP). Although 2-opt is widely used in practice, it is hard to understand its success from a theoretical perspective. We take a statistical approach and examine the features of TSP instances that make the problem either hard or easy to solve. As a measure of problem difficulty for 2-opt we use the approximation ratio that it achieves on a given instance. Our investigations point out important features that make TSP instances hard or easy to be approximated by 2-opt.

On the Properties of the R2 Indicator - Brockhoff, Dimo; Wagner, Tobias; Trautmann, Heike (Proceedings of the 14th Annual Conference on Genetic and Evolutionary Computation)
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.
Keywords: hypervolume indicator; multiobjective optimization; performance assessment; r2 indicator