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Approximating the least hypervolume contributor: NPhard in general, but fast in practice
, 2008
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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 17 (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.
DecompositionBased Memetic Algorithm for Multiobjective Capacitated . . .
 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION
, 2010
"... The capacitated arc routing problem (CARP) is a challenging combinatorial optimization problem with many realworld applications, e.g., salting route optimization and fleet management. There have been many attempts at solving CARP using heuristic and metaheuristic approaches, including evolutionary ..."
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Cited by 14 (3 self)
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The capacitated arc routing problem (CARP) is a challenging combinatorial optimization problem with many realworld applications, e.g., salting route optimization and fleet management. There have been many attempts at solving CARP using heuristic and metaheuristic approaches, including evolutionary algorithms. However, almost all such attempts formulate CARP as a singleobjective problem although it usually has more than one objective, especially considering its realworld applications. This paper studies multiobjective CARP (MOCARP). A new memetic algorithm (MA) called decompositionbased MA with extended neighborhood search (DMAENS) is proposed. The new algorithm combines the advanced features from both the MAENS approach for singleobjective CARP and multiobjective evolutionary optimization. Our experimental studies have shown that such combination outperforms significantly an offtheshelf multiobjective evolutionary algorithm, namely nondominated sorting genetic algorithm II, and the stateoftheart multiobjective algorithm for MOCARP (LMOGA). Our work has also shown that a specifically designed multiobjective algorithm by combining its singleobjective version and multiobjective features may lead to competitive multiobjective algorithms for multiobjective combinatorial optimization problems.
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 (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.
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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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 11 (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 ()
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 10 (4 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
Articulating User Preferences in ManyObjective Problems by Sampling the Weighted Hypervolume
"... The hypervolume indicator has become popular in recent years both for performance assessment and to guide the search of evolutionary multiobjective optimizers. Two critical research topics can be emphasized with respect to hypervolumebased search: (i) the hypervolume indicator inherently introduces ..."
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Cited by 6 (2 self)
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The hypervolume indicator has become popular in recent years both for performance assessment and to guide the search of evolutionary multiobjective optimizers. Two critical research topics can be emphasized with respect to hypervolumebased search: (i) the hypervolume indicator inherently introduces a specific preference and the question is how arbitrary user preferences can be incorporated; (ii) the exact calculation of the hypervolume indicator is expensive and efficient approaches to tackle manyobjective problems are needed. In two previous studies, we addressed both issues independently: a study proposed the weighted hypervolume indicator with which userdefined preferences can be articulated; other studies exist that propose to estimate the hypervolume indicator by MonteCarlo sampling. Here, we combine these two approaches for the first time and extend them, i.e., we present an approach of sampling the weighted hypervolume to incorporate userdefined preferences into the search for problems with many objectives. In particular, we propose weight distribution functions to stress extreme solutions and to define preferred regions of the objective space in terms of socalled preference points; sampling them allows to tackle problems with many objectives. Experiments on several test functions with up to 25 objectives show the usefulness of the approach in terms of decision making and search.
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 5 (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 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.