Multi-Objective Optimization Using Evolutionary Algorithms
Multi-Objective Optimization Using Evolutionary Algorithms
COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
Learning probability distributions in continuous evolutionary algorithms– a comparative review
Natural Computing: an international journal
Completely Derandomized Self-Adaptation in Evolution Strategies
Evolutionary Computation
Multicriteria Optimization
Evolutionary Algorithms for Solving Multi-Objective Problems (Genetic and Evolutionary Computation)
Evolutionary Algorithms for Solving Multi-Objective Problems (Genetic and Evolutionary Computation)
Covariance Matrix Adaptation for Multi-objective Optimization
Evolutionary Computation
The Journal of Machine Learning Research
Approximating the Least Hypervolume Contributor: NP-Hard in General, But Fast in Practice
EMO '09 Proceedings of the 5th International Conference on Evolutionary Multi-Criterion Optimization
S-metric calculation by considering dominated hypervolume as klee's measure problem
Evolutionary Computation
Steady-state selection and efficient covariance matrix update in the multi-objective CMA-ES
EMO'07 Proceedings of the 4th international conference on Evolutionary multi-criterion optimization
Approximating the volume of unions and intersections of high-dimensional geometric objects
Computational Geometry: Theory and Applications
Improved step size adaptation for the MO-CMA-ES
Proceedings of the 12th annual conference on Genetic and evolutionary computation
Comparing results of 31 algorithms from the black-box optimization benchmarking BBOB-2009
Proceedings of the 12th annual conference companion 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
Introduction to Evolutionary Computing
Introduction to Evolutionary Computing
The logarithmic hypervolume indicator
Proceedings of the 11th workshop proceedings on Foundations of genetic algorithms
Hype: An algorithm for fast hypervolume-based many-objective optimization
Evolutionary Computation
EMO'11 Proceedings of the 6th international conference on Evolutionary multi-criterion optimization
An improved algorithm for Klee's measure problem on fat boxes
Computational Geometry: Theory and Applications
Approximating the least hypervolume contributor: NP-hard in general, but fast in practice
Theoretical Computer Science
Multiobjective evolutionary algorithms: a comparative case studyand the strength Pareto approach
IEEE Transactions on Evolutionary Computation
A fast and elitist multiobjective genetic algorithm: NSGA-II
IEEE Transactions on Evolutionary Computation
Performance assessment of multiobjective optimizers: an analysis and review
IEEE Transactions on Evolutionary Computation
A Fast Incremental Hypervolume Algorithm
IEEE Transactions on Evolutionary Computation
On Klee's measure problem for grounded boxes
Proceedings of the twenty-eighth annual symposium on Computational geometry
A Fast Way of Calculating Exact Hypervolumes
IEEE Transactions on Evolutionary Computation
Approximation quality of the hypervolume indicator
Artificial Intelligence
Parameterized average-case complexity of the hypervolume indicator
Proceedings of the 15th annual conference on Genetic and evolutionary computation
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Many state-of-the-art evolutionary vector optimization algorithms compute the contributing hypervolume for ranking candidate solutions. However, with an increasing number of objectives, calculating the volumes becomes intractable. Therefore, although hypervolume-based algorithms are often the method of choice for bi-criteria optimization, they are regarded as not suitable for many-objective optimization. Recently, Monte Carlo methods have been derived and analyzed for approximating the contributing hypervolume. Turning theory into practice, we employ these results in the ranking procedure of the multi-objective covariance matrix adaptation evolution strategy (MO-CMA-ES) as an example of a state-of-the-art method for vector optimization. It is empirically shown that the approximation does not impair the quality of the obtained solutions given a budget of objective function evaluations, while considerably reducing the computation time in the case of multiple objectives. These results are obtained on common benchmark functions as well as on two design optimization tasks. Thus, employing Monte Carlo approximations makes hypervolume-based algorithms applicable to many-objective optimization.