Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Case Study
PPSN V Proceedings of the 5th International Conference on Parallel Problem Solving from Nature
Comparison of Multiobjective Evolutionary Algorithms: Empirical Results
Evolutionary Computation
Covariance Matrix Adaptation for Multi-objective Optimization
Evolutionary Computation
G-Metric: an M-ary quality indicator for the evaluation of non-dominated sets
Proceedings of the 10th annual conference on Genetic and evolutionary computation
An adaptive divide-and-conquer methodology for evolutionary multi-criterion optimisation
EMO'03 Proceedings of the 2nd international conference on Evolutionary multi-criterion optimization
The measure of Pareto optima applications to multi-objective metaheuristics
EMO'03 Proceedings of the 2nd international conference on Evolutionary multi-criterion optimization
Pareto-, aggregation-, and indicator-based methods in many-objective optimization
EMO'07 Proceedings of the 4th international conference on Evolutionary multi-criterion optimization
EMO'07 Proceedings of the 4th international conference on Evolutionary multi-criterion optimization
HM'07 Proceedings of the 4th international conference on Hybrid metaheuristics
An EMO algorithm using the hypervolume measure as selection criterion
EMO'05 Proceedings of the Third international conference on Evolutionary Multi-Criterion Optimization
Properties of an adaptive archiving algorithm for storing nondominated vectors
IEEE Transactions on Evolutionary Computation
Performance assessment of multiobjective optimizers: an analysis and review
IEEE Transactions on Evolutionary Computation
IEEE Transactions on Evolutionary Computation
Articulating user preferences in many-objective problems by sampling the weighted hypervolume
Proceedings of the 11th Annual conference on Genetic and evolutionary computation
Investigating and exploiting the bias of the weighted hypervolume to articulate user preferences
Proceedings of the 11th Annual conference on Genetic and evolutionary computation
Multiplicative approximations and the hypervolume indicator
Proceedings of the 11th Annual conference on Genetic and evolutionary computation
Single-objective and multi-objective formulations of solution selection for hypervolume maximization
Proceedings of the 11th Annual conference on Genetic and evolutionary computation
A mono surrogate for multiobjective optimization
Proceedings of the 12th annual conference on Genetic and evolutionary computation
The maximum hypervolume set yields near-optimal approximation
Proceedings of the 12th annual conference on Genetic and evolutionary computation
Simultaneous use of different scalarizing functions in MOEA/D
Proceedings of the 12th annual conference on Genetic and evolutionary computation
PPSN'10 Proceedings of the 11th international conference on Parallel problem solving from nature: Part I
Tight bounds for the approximation ratio of the hypervolume indicator
PPSN'10 Proceedings of the 11th international conference on Parallel problem solving from nature: Part I
On expected-improvement criteria for model-based multi-objective optimization
PPSN'10 Proceedings of the 11th international conference on Parallel problem solving from nature: Part I
PPSN'10 Proceedings of the 11th international conference on Parallel problem solving from nature: Part II
IEEE Transactions on Evolutionary Computation - Special issue on preference-based multiobjective evolutionary algorithms
SEAL'10 Proceedings of the 8th international conference on Simulated evolution and learning
Dominance-based pareto-surrogate for multi-objective optimization
SEAL'10 Proceedings of the 8th international conference on Simulated evolution and learning
Illustration of fairness in evolutionary multi-objective optimization
Theoretical Computer Science
The logarithmic hypervolume indicator
Proceedings of the 11th workshop proceedings on Foundations of genetic algorithms
Not all parents are equal for MO-CMA-ES
EMO'11 Proceedings of the 6th international conference on Evolutionary multi-criterion optimization
Information Sciences: an International Journal
A performance study on synchronous and asynchronous updates in particle swarm optimization
Proceedings of the 13th annual conference on Genetic and evolutionary computation
Proceedings of the 13th annual conference on Genetic and evolutionary computation
Hypervolume-based multiobjective optimization: Theoretical foundations and practical implications
Theoretical Computer Science
Fast approximation heuristics for multi-objective vehicle routing problems
EvoCOMNET'10 Proceedings of the 2010 international conference on Applications of Evolutionary Computation - Volume Part II
On the properties of the R2 indicator
Proceedings of the 14th annual conference on Genetic and evolutionary computation
GECCO 2012 tutorial on evolutionary multiobjective optimization
Proceedings of the 14th annual conference companion on Genetic and evolutionary computation
Recombination of similar parents in SMS-EMOA on many-objective 0/1 knapsack problems
PPSN'12 Proceedings of the 12th international conference on Parallel Problem Solving from Nature - Volume Part II
Approximation quality of the hypervolume indicator
Artificial Intelligence
International Journal of Swarm Intelligence Research
A fast approximation-guided evolutionary multi-objective algorithm
Proceedings of the 15th annual conference on Genetic and evolutionary computation
GECCO 2013 tutorial on evolutionary multiobjective optimization
Proceedings of the 15th annual conference companion on Genetic and evolutionary computation
Asynchronous master-slave parallelization of differential evolution for multi-objective optimization
Evolutionary Computation
Computational Intelligence and Neuroscience
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The hypervolume indicator is a set measure used in evolutionary multiobjective optimization to evaluate the performance of search algorithms and to guide the search. Multiobjective evolutionary algorithms using the hypervolume indicator transform multiobjective problems into single objective ones by searching for a finite set of solutions maximizing the corresponding hypervolume indicator. In this paper, we theoretically investigate how those optimal μ--distributions-finite sets of μ solutions maximizing the hypervolume indicator-are spread over the Pareto front of biobjective problems. This problem is of high importance for practical applications as these sets characterize the preferences that the hypervolume indicator encodes, i.e., which types of Pareto set approximations are favored. In particular, we tackle the question whether the hypervolume indicator is biased towards certain regions. For linear fronts we prove that the distribution is uniform with constant distance between two consecutive points. For general fronts where it is presumably impossible to characterize exactly the distribution, we derive a limit result when the number of points grows to infinity proving that the empirical density of points converges to a density proportional to the square root of the negative of the derivative of the front. Our analyses show that it is not the shape of the Pareto front but only its slope that determines how the points that maximize the hypervolume indicator are distributed. Experimental results illustrate that the limit density is a good approximation of the empirical density for small μ. Furthermore, we analyze the issue of where to place the reference point of the indicator such that the extremes of the front can be found if the hypervolume indicator is optimized. We derive an explicit lower bound (possibly infinite) ensuring the presence of the extremes in the optimal distribution. This result contradicts the common belief that the reference point has to be chosen close to the nadir point: for certain types of fronts, we show that no finite reference point allows to have the extremes in the optimal μ-distribution.