New upper bounds in Klee's measure problem
SIAM Journal on Computing
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 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
Approximating the Volume of Unions and Intersections of High-Dimensional Geometric Objects
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
EMO'07 Proceedings of the 4th international conference on Evolutionary multi-criterion optimization
An EMO algorithm using the hypervolume measure as selection criterion
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
Proceedings of the 12th annual conference companion on Genetic and evolutionary computation
An efficient algorithm for computing hypervolume contributions**
Evolutionary Computation
Illustration of fairness in evolutionary multi-objective optimization
Theoretical Computer Science
On sequential online archiving of objective vectors
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
Expert Systems with Applications: An International Journal
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Most hypervolume indicator based optimization algorithms like SIBEA [Zitzler et al. 2007], SMS-EMOA [Beume et al. 2007], or MO-CMA-ES [Igel et al. 2007] remove the solution with the smallest loss with respect to the dominated hypervolume from the population. This is usually iterated λ times until the size of the population no longer exceeds a fixed size μ. We show that there are populations such that the contributing hypervolume of the λ solutions chosen by this greedy selection scheme can be much smaller than the contributing hypervolume of an optimal set of λ solutions. Selecting the optimal λ-set implies calculating (μ+λ over μ)conventional hypervolume contributions, which is considered computationally too expensive. We present the first hypervolume algorithm which calculates directly the contribution of every set of λ solutions. This gives an additive term of (μ+λ over μ)in the runtime of the calculation instead of a multiplicative factor of binomial(μ+λ over μ). Given a population of size n=μ+λ, our algorithm can calculate a set of λ≥ solutions with minimal d-dimensional hypervolume contribution in time O(nd/2 log + n λ) for d 2. This improves all previously published algorithms by a factor of order nmin(λ,,d/2 for d 3. Therefore even if we remove the solutions one by one greedily as usual, we gain a speedup factor of n for all d 3.