Real and complex analysis, 3rd ed.
Real and complex analysis, 3rd ed.
Combining convergence and diversity in evolutionary multiobjective optimization
Evolutionary Computation
On the approximability of trade-offs and optimal access of Web sources
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
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
Theory of the hypervolume indicator: optimal μ-distributions and the choice of the reference point
Proceedings of the tenth ACM SIGEVO workshop on Foundations of genetic algorithms
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
EMO'07 Proceedings of the 4th international conference on Evolutionary multi-criterion optimization
The maximum hypervolume set yields near-optimal approximation
Proceedings of the 12th annual conference on Genetic and evolutionary computation
An EMO algorithm using the hypervolume measure as selection criterion
EMO'05 Proceedings of the Third international conference on Evolutionary Multi-Criterion Optimization
Running time analysis of a multiobjective evolutionary algorithm on simple and hard problems
FOGA'05 Proceedings of the 8th international conference on Foundations of Genetic Algorithms
Multiobjective evolutionary algorithms: a comparative case studyand the strength Pareto approach
IEEE Transactions on Evolutionary Computation
Properties of an adaptive archiving algorithm for storing nondominated vectors
IEEE Transactions on Evolutionary Computation
The logarithmic hypervolume indicator
Proceedings of the 11th workshop proceedings on Foundations of genetic algorithms
Approximation-guided evolutionary multi-objective optimization
IJCAI'11 Proceedings of the Twenty-Second international joint conference on Artificial Intelligence - Volume Volume Two
Convergence of set-based multi-objective optimization, indicators and deteriorative cycles
Theoretical Computer Science
Approximation quality of the hypervolume indicator
Artificial Intelligence
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The hypervolume indicator is widely used to guide the search and to evaluate the performance of evolutionary multi-objective 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 two-dimensional 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.