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
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
The measure of Pareto optima applications to multi-objective metaheuristics
EMO'03 Proceedings of the 2nd international conference on Evolutionary multi-criterion optimization
Pareto-, aggregation-, and indicator-based methods in many-objective 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
Hypervolume-based multiobjective optimization: Theoretical foundations and practical implications
Theoretical Computer Science
Performance assessment of multiobjective optimizers: an analysis and review
IEEE Transactions on Evolutionary Computation
A review of multiobjective test problems and a scalable test problem toolkit
IEEE Transactions on Evolutionary Computation
Hypervolume-based multiobjective optimization: Theoretical foundations and practical implications
Theoretical Computer Science
Population size matters: rigorous runtime results for maximizing the hypervolume indicator
Proceedings of the 15th annual conference on Genetic and evolutionary computation
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To simultaneously optimize multiple objective functions, several evolutionary multiobjective optimization (EMO) algorithms have been proposed. Nowadays, often set quality indicators are used when comparing the performance of those algorithms or when selecting "good" solutions during the algorithm run. Hence, characterizing the solution sets that maximize a certain indicator is crucial--complying with the optimization goal of many indicator-based EMO algorithms. If these optimal solution sets are upper bounded in size, e.g., by the population size µ, we call them optimal µ-distributions. Recently, optimal µ-distributions for the well-known hypervolume indicator have been theoretically analyzed, in particular, for bi-objective problems with a linear Pareto front. Although the exact optimal µ-distributions have been characterized in this case, not all possible choices of the hypervolume's reference point have been investigated. In this paper, we revisit the previous results and rigorously characterize the optimal µ-distributions also for all other reference point choices. In this sense, our characterization is now exhaustive as the result holds for any linear Pareto front and for any choice of the reference point and the optimal µ-distributions turn out to be always unique in those cases. We also prove a tight lower bound (depending on µ) such that choosing the reference point above this bound ensures the extremes of the Pareto front to be always included in optimal µ-distributions.