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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 14 (4 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.
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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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
Tight bounds for the approximation ratio of the hypervolume indicator
 In Proc. 11th International Conference 29 Problem Solving from Nature (PPSN XI), volume 6238 of LNCS
, 2010
"... Abstract The hypervolume indicator is widely used to guide the search and to evaluate the performance of evolutionary multiobjective optimization algorithms. It measures the volume of the dominated portion of the objective space which is considered to give a good approximation of the Pareto front. ..."
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Abstract The hypervolume indicator is widely used to guide the search and to evaluate the performance of evolutionary multiobjective optimization algorithms. It measures the volume of the dominated portion of the objective space which is considered to give a good approximation of the Pareto front. There is surprisingly little theoretically known about the quality of this approximation. We examine the multiplicative approximation ratio achieved by twodimensional sets maximizing the hypervolume indicator and prove that it deviates significantly from the optimal approximation ratio. This provable gap is even exponential in the ratio between the largest and the smallest value of the front. We also examine the additive approximation ratio of the hypervolume indicator and prove that it achieves the optimal additive approximation ratio apart from a small factor � n/(n − 2), where n is the size of the population. Hence the hypervolume indicator can be used to achieve a very good additive but not a good multiplicative approximation of a Pareto front. 1
The Logarithmic Hypervolume Indicator
, 2011
"... It was recently proven that sets of points maximizing the hypervolume indicator do not give a good multiplicative approximation of the Pareto front. We introduce a new“logarithmic hypervolume indicator”and prove that it achieves a closetooptimal multiplicative approximation ratio. This is experime ..."
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It was recently proven that sets of points maximizing the hypervolume indicator do not give a good multiplicative approximation of the Pareto front. We introduce a new“logarithmic hypervolume indicator”and prove that it achieves a closetooptimal multiplicative approximation ratio. This is experimentally verified on several benchmark functions by comparing the approximation quality of the multiobjective covariance matrix evolution strategy (MOCMAES) with the classic hypervolume indicator and the MOCMAES with the logarithmic hypervolume indicator.
Approximation Quality of the Hypervolume Indicator
, 2012
"... 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 4 (3 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 bounded from below by a reference point. Though the hypervolume indicator is very popular, it has not been shown that maximizing the hypervolume indicator of sets of bounded size 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 ratio with the approximation ratio achieved by twodimensional sets maximizing the hypervolume indicator. We bound the optimal multiplicative approximation ratio 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. However, there is a provable gap between the two approximation ratios which is even exponential in the ratio between the largest and the smallest value of the front. We also examine the additive approximation ratio of the hypervolume indicator in two dimensions and prove that it achieves the optimal additive approximation ratio apart from a small ratio � n/(n−2), where n is the size of the population. Hence the hypervolume indicator can be used to achieve a good additive but not a good multiplicative approximation of a Pareto front. This motivates the introduction of a “logarithmic hypervolume indicator ” which provably achieves a good multiplicative approximation ratio.
Simultaneous Use of Different Scalarizing Functions in MOEA/D
"... The use of Pareto dominance for fitness evaluation has been the mainstream in evolutionary multiobjective optimization for the last two decades. Recently, it has been pointed out in some studies that Pareto dominancebased algorithms do not always work well on multiobjective problems with many objec ..."
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The use of Pareto dominance for fitness evaluation has been the mainstream in evolutionary multiobjective optimization for the last two decades. Recently, it has been pointed out in some studies that Pareto dominancebased algorithms do not always work well on multiobjective problems with many objectives. Scalarizing functionbased fitness evaluation is a promising alternative to Pareto dominance especially for the case of many objectives. A representative scalarizing functionbased algorithm is MOEA/D (multiobjective evolutionary algorithm based on decomposition) of Zhang & Li (2007). Its high search ability has already been shown for various problems. One important implementation issue of MOEA/D is a choice of a scalarizing function because its search ability strongly depends on this choice. It is, however, not easy to choose an appropriate scalarizing function for each multiobjective problem. In this paper, we propose an idea of using different types of scalarizing functions simultaneously. For example, both the weighted Tchebycheff (Chebyshev) and the weighted sum are used for fitness evaluation. We examine two methods for implementing our idea. One is to use multiple grids of weight vectors and the other is to assign a different scalarizing function alternately to each weight vector in a single grid.
A Fast ApproximationGuided Evolutionary MultiObjective Algorithm
"... ApproximationGuided Evolution (AGE) [4] is a recently presented multiobjective algorithm that outperforms stateoftheart multimultiobjective algorithms in terms of approximation quality. This holds for problems with many objectives, but AGE’s performance is not competitive on problems with few ..."
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ApproximationGuided Evolution (AGE) [4] is a recently presented multiobjective algorithm that outperforms stateoftheart multimultiobjective algorithms in terms of approximation quality. This holds for problems with many objectives, but AGE’s performance is not competitive on problems with few objectives. Furthermore, AGE is storing all nondominated points seen so far in an archive, which can have very detrimental effects on its runtime. In this article, we present the fast approximationguided evolutionary algorithm called AGEII. It approximates the archive in order to control its size and its influence on the runtime. This allows for tradingoff approximation and runtime, and it enables a faster approximation process. Our experiments show that AGEII performs very well for multiobjective problems having few as well as many objectives. It scales well with the number of objectives and enables practitioners to add objectives to their problems at small additional computational cost.
ManyObjective Test Problems to Visually Examine the Behavior of Multiobjective Evolution in a Decision Space
"... Abstract. Manyobjective optimization is a hot issue in the EMO (evolutionary multiobjective optimization) community. Since almost all solutions in the current population are nondominated with each other in manyobjective EMO algorithms, we may need a different fitness evaluation scheme from the ca ..."
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Abstract. Manyobjective optimization is a hot issue in the EMO (evolutionary multiobjective optimization) community. Since almost all solutions in the current population are nondominated with each other in manyobjective EMO algorithms, we may need a different fitness evaluation scheme from the case of two and three objectives. One difficulty in the design of manyobjective EMO algorithms is that we cannot visually observe the behavior of multiobjective evolution in the objective space with four or more objectives. In this paper, we propose the use of manyobjective test problems in a two or threedimensional decision space to visually examine the behavior of multiobjective evolution. Such a visual examination helps us to understand the characteristic features of EMO algorithms for manyobjective optimization. Good understanding of existing EMO algorithms may facilitates their modification and the development of new EMO algorithms for manyobjective optimization.
Optimal µDistributions for the Hypervolume Indicator for Problems With Linear BiObjective Fronts: Exact and Exhaustive Results, in "Simulated Evolution And Learning (SEAL2010)", Inde Kanpur, November 2010, corrected author version
"... Abstract. To simultaneously optimize multiple objective functions, several evolutionary multiobjective optimization (EMO) algorithms have been proposed. Nowadays, often set quality indicators are used when comparing the performance of those algorithms or when selecting “good ” solutions during the ..."
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Abstract. To simultaneously optimize multiple objective functions, several evolutionary multiobjective optimization (EMO) algorithms have been proposed. Nowadays, often set quality indicators are used when comparing the performance of those algorithms or when selecting “good ” solutions during the algorithm run. Hence, characterizing the solution sets that maximize a certain indicator is crucial—complying with the optimization goal of many indicatorbased EMO algorithms. If these optimal solution sets are upper bounded in size, e.g., by the population size μ, we call them optimal μdistributions. Recently, optimal μdistributions for the wellknown hypervolume indicator have been theoretically analyzed, in particular, for biobjective problems with a linear Pareto front. Although the exact optimal μdistributions have been characterized in this case, not all possible choices of the hypervolume’s reference point have been investigated. In this paper, we revisit the previous results and rigorously characterize the optimal μdistributions also for all other reference point choices. In this sense, our characterization is now exhaustive as the result holds for any linear Pareto front and for any choice of the reference point and the optimal μdistributions turn out to be always unique in those cases. We also prove a tight lower bound (depending on μ) such that choosing the reference point above this bound ensures the extremes of the Pareto front to be always included in optimal μdistributions.
Theoretically Investigating Optimal µDistributions for the Hypervolume Indicator: First Results For Three Objectives, in "Parallel Problem Solving from Nature
 22 Activity Report INRIA 2010
"... Abstract. Several indicatorbased evolutionary multiobjective optimization algorithms have been proposed in the literature. The notion of optimal µdistributions formalizes the optimization goal of such algorithms: find a set of µ solutions that maximizes the underlying indicator among all sets wi ..."
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Abstract. Several indicatorbased evolutionary multiobjective optimization algorithms have been proposed in the literature. The notion of optimal µdistributions formalizes the optimization goal of such algorithms: find a set of µ solutions that maximizes the underlying indicator among all sets with µ solutions. In particular for the often used hypervolume indicator, optimal µdistributions have been theoretically analyzed recently. All those results, however, cope with biobjective problems only. It is the main goal of this paper to extend some of the results to the 3objective case. This generalization is shown to be not straightforward as a solution’s hypervolume contribution has not a simple geometric shape anymore in opposition to the biobjective case where it is always rectangular. In addition, we investigate the influence of the reference point on optimal µdistributions and prove that also in the 3objective case situations exist for which the Pareto front’s extreme points cannot be guaranteed in optimal µdistributions. 1