Multiobjective evolutionary algorithm test suites
Proceedings of the 1999 ACM symposium on Applied computing
Covariance Matrix Adaptation for Multi-objective Optimization
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
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
HM'07 Proceedings of the 4th international conference on Hybrid metaheuristics
On set-based multiobjective optimization
IEEE Transactions on Evolutionary Computation
The maximum hypervolume set yields near-optimal approximation
Proceedings of the 12th annual conference on Genetic and evolutionary computation
Multiobjective evolutionary algorithms: a comparative case studyand the strength Pareto approach
IEEE Transactions on Evolutionary Computation
A review of multiobjective test problems and a scalable test problem toolkit
IEEE Transactions on Evolutionary Computation
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Several indicator-based evolutionary multiobjective optimization algorithms have been proposed in the literature. The notion of optimal µ-distributions formalizes the optimization goal of such algorithms: find a set of µ solutions that maximizes the underlying indicator among all sets with µ solutions. In particular for the often used hypervolume indicator, optimal µ-distributions have been theoretically analyzed recently. All those results, however, cope with bi-objective problems only. It is the main goal of this paper to extend some of the results to the 3-objective case. This generalization is shown to be not straight-forward as a solution's hypervolume contribution has not a simple geometric shape anymore in opposition to the bi-objective case where it is always rectangular. In addition, we investigate the influence of the reference point on optimal µ-distributions and prove that also in the 3-objective case situations exist for which the Pareto front's extreme points cannot be guaranteed in optimal µ-distributions.