Real and complex analysis, 3rd ed.
Real and complex analysis, 3rd ed.
Multiobjective query optimization
PODS '01 Proceedings of the twentieth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Bicriterion Single Machine Scheduling with Resource Dependent Processing Times
SIAM Journal on Optimization
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
Efficiently computing succinct trade-off curves
Theoretical Computer Science - Automata, languages and programming: Algorithms and complexity (ICALP-A 2004)
Covariance Matrix Adaptation for Multi-objective Optimization
Evolutionary Computation
A Fast Algorithm for Computing the Contribution of a Point to the Hypervolume
ICNC '07 Proceedings of the Third International Conference on Natural Computation - Volume 04
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
Approximating the Volume of Unions and Intersections of High-Dimensional Geometric Objects
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and 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
S-metric calculation by considering dominated hypervolume as klee's measure problem
Evolutionary Computation
EMO'07 Proceedings of the 4th international conference on Evolutionary multi-criterion optimization
How good is the Chord algorithm?
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Small Approximate Pareto Sets for Biobjective Shortest Paths and Other Problems
SIAM Journal on Computing
Hype: An algorithm for fast hypervolume-based many-objective optimization
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
Performance assessment of multiobjective optimizers: an analysis and review
IEEE Transactions on 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
SEAL'10 Proceedings of the 8th international conference on Simulated evolution and learning
The logarithmic hypervolume indicator
Proceedings of the 11th workshop proceedings on Foundations of genetic algorithms
On sequential online archiving of objective vectors
EMO'11 Proceedings of the 6th international conference on Evolutionary multi-criterion optimization
Hypervolume-based multiobjective optimization: Theoretical foundations and practical implications
Theoretical Computer Science
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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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 near-optimal approximation ratio.