New upper bounds in Klee's measure problem
SIAM Journal on Computing
On the complexity of computing the measure of ∪[ai,bi]
Communications of the ACM
Multi-Objective Optimization Using Evolutionary Algorithms
Multi-Objective Optimization Using Evolutionary Algorithms
Evolutionary Algorithms for Solving Multi-Objective Problems
Evolutionary Algorithms for Solving Multi-Objective Problems
Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Case Study
PPSN V Proceedings of the 5th International Conference on Parallel Problem Solving from Nature
Covariance Matrix Adaptation for Multi-objective Optimization
Evolutionary Computation
A (slightly) faster algorithm for klee's measure problem
Proceedings of the twenty-fourth annual symposium on Computational geometry
Approximating the Volume of Unions and Intersections of High-Dimensional Geometric Objects
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
On the complexity of computing the hypervolume indicator
IEEE Transactions on Evolutionary Computation
The measure of Pareto optima applications to multi-objective metaheuristics
EMO'03 Proceedings of the 2nd international conference on Evolutionary multi-criterion optimization
EMO'07 Proceedings of the 4th international conference on Evolutionary multi-criterion optimization
A new analysis of the lebmeasure algorithm for calculating hypervolume
EMO'05 Proceedings of the Third international conference on Evolutionary Multi-Criterion Optimization
Performance assessment of multiobjective optimizers: an analysis and review
IEEE Transactions on Evolutionary Computation
A faster algorithm for calculating hypervolume
IEEE Transactions on Evolutionary Computation
A Fast Incremental Hypervolume Algorithm
IEEE Transactions on Evolutionary Computation
Statistical methods for convergence detection of multi-objective evolutionary algorithms
Evolutionary Computation
Improved step size adaptation for the MO-CMA-ES
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
Proceedings of the 12th annual conference on Genetic and evolutionary computation
Proceedings of the 12th annual conference companion on Genetic and evolutionary computation
An efficient algorithm for computing hypervolume contributions**
Evolutionary Computation
EMO'11 Proceedings of the 6th international conference on Evolutionary multi-criterion optimization
Achieving balance between proximity and diversity in multi-objective evolutionary algorithm
Information Sciences: an International Journal
Approximating the least hypervolume contributor: NP-hard in general, but fast in practice
Theoretical Computer Science
LION'05 Proceedings of the 5th international conference on Learning and Intelligent Optimization
Parameterized average-case complexity of the hypervolume indicator
Proceedings of the 15th annual conference on Genetic and evolutionary computation
Speeding up many-objective optimization by Monte Carlo approximations
Artificial Intelligence
Annals of Mathematics and Artificial Intelligence
| Hi-index | 0.00 |
The dominated hypervolume (or S-metric) is a commonly accepted quality measure for comparing approximations of Pareto fronts generated by multi-objective optimizers. Since optimizers exist, namely evolutionary algorithms, that use the S-metric internally several times per iteration, a fast determination of the S-metric value is of essential importance. This work describes how to consider the S-metric as a special case of a more general geometric problem called Klee's measure problem (KMP). For KMP, an algorithm exists with runtime O(n log n + nd/2 log n), for n points of d ≥ 3 dimensions. This complex algorithm is adapted to the special case of calculating the S-metric. Conceptual simplifications realize the algorithm without complex data structures and establish an upper bound of O(nd/2 log n) for the S-metric calculation for d ≥ 3. The performance of the new algorithm is studied in comparison to another state of the art algorithm on a set of academic test functions.