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HypE: An Algorithm for Fast HypervolumeBased ManyObjective Optimization
, 2008
"... In the field of evolutionary multicriterion optimization, the hypervolume indicator is the only single set quality measure that is known to be strictly monotonic with regard to Pareto dominance: whenever a Pareto set approximation entirely dominates another one, then also the indicator value of the ..."
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Cited by 53 (5 self)
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In the field of evolutionary multicriterion optimization, the hypervolume indicator is the only single set quality measure that is known to be strictly monotonic with regard to Pareto dominance: whenever a Pareto set approximation entirely dominates another one, then also the indicator value of the former will be better. This property is of high interest and relevance for problems involving a large number of objective functions. However, the high computational effort required for hypervolume calculation has so far prevented to fully exploit the potential of this indicator; current hypervolumebased search algorithms are limited to problems with only a few objectives. This paper addresses this issue and proposes a fast search algorithm that uses Monte Carlo simulation to approximate the exact hypervolume values. The main idea is that not the actual indicator values are important, but rather the rankings of solutions induced by the hypervolume indicator. In detail, we present HypE, a hypervolume estimation algorithm for multiobjective optimization, by which the accuracy of the estimates and the available computing resources can be traded off; thereby, not only manyobjective problems become feasible with hypervolumebased search, but also the runtime can be flexibly adapted. Moreover, we show how the same principle can be used to statistically compare the outcomes of different multiobjective optimizers with respect to the hypervolume—so far, statistical testing has been restricted to scenarios with few objectives. The experimental results indicate that HypE is highly effective for manyobjective problems in comparison to existing multiobjective evolutionary algorithms. HypE is available for download at
On SetBased Multiobjective Optimization
, 2008
"... Assuming that evolutionary multiobjective optimization (EMO) mainly deals with set problems, one can identify three core questions in this area of research: (i) how to formalize what type of Pareto set approximation is sought, (ii) how to use this information within an algorithm to efficiently sear ..."
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Cited by 25 (5 self)
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Assuming that evolutionary multiobjective optimization (EMO) mainly deals with set problems, one can identify three core questions in this area of research: (i) how to formalize what type of Pareto set approximation is sought, (ii) how to use this information within an algorithm to efficiently search for a good Pareto set approximation, and (iii) how to compare the Pareto set approximations generated by different optimizers with respect to the formalized optimization goal. There is a vast amount of studies addressing these issues from different angles, but so far only few studies can be found that consider all questions under one roof. This paper is an attempt to summarize recent developments in the EMO field within a unifying theory of setbased multiobjective search. It discusses how preference relations on sets can be formally defined, gives examples for selected user preferences, and proposes a general, preferenceindependent hill climber for multiobjective optimization with theoretical convergence properties. Furthermore, it shows how to use set preference relations for statistical performance assessment and provides corresponding experimental results. The proposed methodology brings together preference articulation, algorithm design, and performance assessment under one framework and thereby opens up a new perspective on EMO.
Multiplicative Approximations and the Hypervolume Indicator
"... Indicatorbased algorithms have become a very popular approach to solve multiobjective optimization problems. In this paper, we contribute to the theoretical understanding of algorithms maximizing the hypervolume for a given problem by distributing µ points on the Pareto front. We examine this comm ..."
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Cited by 19 (7 self)
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Indicatorbased algorithms have become a very popular approach to solve multiobjective optimization problems. In this paper, we contribute to the theoretical understanding of algorithms maximizing the hypervolume for a given problem by distributing µ points on the Pareto front. We examine this common approach with respect to the achieved multiplicative approximation ratio for a given multiobjective problem and relate it to a set of µ points on the Pareto front that achieves the best possible approximation ratio. For the class of linear fronts and a class of concave fronts, we prove that the hypervolume gives the best possible approximation ratio. In addition, we examine Pareto fronts of different shapes by numerical calculations and show that the approximation computed by the hypervolume may differ from the optimal approximation ratio.
Investigating and Exploiting the Bias of the Weighted Hypervolume to Articulate User Preferences
"... Optimizing the hypervolume indicator within evolutionary multiobjective optimizers has become popular in the last years. Recently, the indicator has been generalized to the weighted case to incorporate various user preferences into hypervolumebased search algorithms. There are two main open questio ..."
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Cited by 18 (7 self)
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Optimizing the hypervolume indicator within evolutionary multiobjective optimizers has become popular in the last years. Recently, the indicator has been generalized to the weighted case to incorporate various user preferences into hypervolumebased search algorithms. There are two main open questions in this context: (i) how does the specified weight influence the distribution of a fixed number of points that maximize the weighted hypervolume indicator? (ii) how can the user articulate her preferences easily without specifying a certain weight distribution function? In this paper, we tackle both questions. First, we theoretically investigate optimal distributions of μ points that maximize the weighted hypervolume indicator. Second, based on the obtained theoretical results, we propose a new approach to articulate user preferences within biobjective hypervolumebased optimization in terms of specifying a desired density of points on a predefined (imaginary) Pareto front. Within this approach, a new exact algorithm based on dynamic programming is proposed which selects the set of μ points that maximizes the (weighted) hypervolume indicator. Experiments on various test functions show the usefulness of this new preference articulation approach and the agreement between theory and practice.
The maximum hypervolume set yields nearoptimal approximation
 IN PROC. 12TH ANNUAL CONFERENCE ON GENETIC AND EVOLUTIONARY COMPUTATION (GECCO ’10
, 2010
"... In order to allow a comparison of (otherwise incomparable) sets, many evolutionary multiobjective optimizers use indicator functions to guide the search and to evaluate the performance of search algorithms. The most widely used indicator is the hypervolume indicator. It measures the volume of the do ..."
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Cited by 16 (5 self)
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In order to allow a comparison of (otherwise incomparable) sets, many evolutionary multiobjective optimizers use indicator functions to guide the search and to evaluate the performance of search algorithms. The most widely used indicator is the hypervolume indicator. It measures the volume of the dominated portion of the objective space. Though the hypervolume indicator is very popular, it has not been shown that maximizing the hypervolume indicator is indeed equivalent to the overall objective of finding a good approximation of the Pareto front. To address this question, we compare the optimal approximation factor with the approximation factor achieved by sets maximizing the hypervolume indicator. We bound the optimal approximation factor of n points by 1 + Θ(1/n) for arbitrary Pareto fronts. Furthermore, we prove that the same asymptotic approximation ratio is achieved by sets of n points that maximize the hypervolume indicator. This shows that the speed of convergence of the approximation ratio achieved by maximizing the hypervolume indicator is asymptotically optimal. This implies that for large values of n, sets maximizing the hypervolume indicator quickly approach the optimal approximation ratio. Moreover, our bounds show that also for relatively small values of n, sets maximizing the hypervolume indicator achieve a nearoptimal approximation ratio.
An efficient algorithm for computing hypervolume contributions
 Evolutionary Computation
"... The hypervolume indicator serves as a sorting criterion in many recent multiobjective evolutionary algorithms (MOEAs). Typical algorithms remove the solution with the smallest loss with respect to the dominated hypervolume from the population. We present a new algorithm which determines for a popul ..."
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Cited by 12 (6 self)
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The hypervolume indicator serves as a sorting criterion in many recent multiobjective evolutionary algorithms (MOEAs). Typical algorithms remove the solution with the smallest loss with respect to the dominated hypervolume from the population. We present a new algorithm which determines for a population of size n with d objectives, a solution with minimal hypervolume contribution in time O(n d/2 log n) for d> 2. This improves all previously published algorithms by a factor of n for all d> 3 and by a factor of √ n for d = 3. We also analyze hypervolume indicator based optimization algorithms which remove λ> 1 solutions from a population of size n = µ + λ. We show that there are populations such that the hypervolume contribution of iteratively chosen λ solutions is much larger than the hypervolume contribution of an optimal set of λ solutions. Selecting the optimal set of λ solutions implies calculating () n conventional hypervolume conµ tributions, which is considered to be computationally too expensive. We present the first hypervolume algorithm which calculates directly the contribution of every set of λ solutions. This gives an additive term of () n in the runtime of the calculation inµ stead of a multiplicative factor of ()
Don’t be greedy when calculating hypervolume contributions
 Proceedings of the 10th International Workshop on Foundations of Genetic Algorithms (FOGA 2009
, 2009
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Hypervolumebased Multiobjective Optimization: Theoretical Foundations and Practical Implications
 THEORETICAL COMPUTER SCIENCE
, 2011
"... In recent years, indicatorbased evolutionary algorithms, allowing to implicitly incorporate user preferences into the search, have become widely used in practice to solve multiobjective optimization problems. When using this type of methods, the optimization goal changes from optimizing a set of ob ..."
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Cited by 11 (5 self)
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In recent years, indicatorbased evolutionary algorithms, allowing to implicitly incorporate user preferences into the search, have become widely used in practice to solve multiobjective optimization problems. When using this type of methods, the optimization goal changes from optimizing a set of objective functions simultaneously to the singleobjective optimization goal of finding a set of µ points that maximizes the underlying indicator. Understanding the difference between these two optimization goals is fundamental when applying indicatorbased algorithms in practice. On the one hand, a characterization of the inherent optimization goal of different indicators allows the user to choose the indicator that meets her preferences. On the other hand, knowledge about the sets of µ points with optimal indicator values—socalled optimal µdistributions—can be used in performance assessment whenever the indicator is used as a performance criterion. However, theoretical studies on indicatorbased optimization are sparse. One of the most popular indicators is the weighted hypervolume indicator. It allows to guide the search towards userdefined objective space regions and at the same time has the property of being a refinement of the Pareto dominance relation with the result that maximizing the indicator results in Paretooptimal solutions only. In previous work, we theoretically investigated the unweighted hypervolume indicator in terms of a characterization of optimal µdistributions and the influence of the hypervolume’s reference point for general biobjective optimization problems. In this
Gradientbased/Evolutionary Relay Hybrid for Computing Pareto Front Approximations Maximizing the SMetric
, 2007
"... The computation of a good approximation set of the Pareto front of a multiobjective optimization problem can be recasted as the maximization of its Smetric value. A highprecision method for computing approximation sets of a Pareto front with maximal SMetric is presented in this paper as a highl ..."
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Cited by 10 (1 self)
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The computation of a good approximation set of the Pareto front of a multiobjective optimization problem can be recasted as the maximization of its Smetric value. A highprecision method for computing approximation sets of a Pareto front with maximal SMetric is presented in this paper as a highlevel relay hybrid of an evolutionary algorithm and a gradient method, both guided by the Smetric. First, an evolutionary multiobjective optimizer moves the initial population close to the Pareto front. The gradientbased method takes this population as its starting point for computing a local maximal approximation set with respect to the Smetric. As opposed to existing work on gradientbased multicriteria optimization in the new gradient approach we compute gradients based on a set of points rather than for single points. We will term this approach indicatorbased gradient method, and exemplify it for the Smetric. We derive expressions for computing the gradient of a set of points with respect
R.: Evolving multimodal networks for multitask games
 In: Proceedings of the IEEE Conference on Computational Intelligence and Games (CIG 2011
, 2011
"... Abstract—Intelligent opponent behavior helps make video games interesting to human players. Evolutionary computation can discover such behavior, especially when the game consists of a single task. However, multitask domains, in which separate tasks within the domain each have their own dynamics and ..."
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Cited by 9 (4 self)
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Abstract—Intelligent opponent behavior helps make video games interesting to human players. Evolutionary computation can discover such behavior, especially when the game consists of a single task. However, multitask domains, in which separate tasks within the domain each have their own dynamics and objectives, can be challenging for evolution. This paper proposes two methods for meeting this challenge by evolving neural networks: 1) Multitask Learning provides a network with distinct outputs per task, thus evolving a separate policy for each task, and 2) Mode Mutation provides a means to evolve new output modes, as well as a way to select which mode to use at each moment. Multitask Learning assumes agents know which task they are currently facing; if such information is available and accurate, this approach works very well, as demonstrated in the Front/Back Ramming game of this paper. In contrast, Mode Mutation discovers an appropriate task division on its own, which may in some cases be even more powerful than a humanspecified task division, as shown in the Predator/Prey game of this paper. These results demonstrate the importance of both Multitask Learning and Mode Mutation for learning intelligent behavior in complex games. I.