Multi-Objective Optimization Using Evolutionary Algorithms
Multi-Objective Optimization Using Evolutionary Algorithms
Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Case Study
PPSN V Proceedings of the 5th International Conference on Parallel Problem Solving from Nature
Comparison of Multiobjective Evolutionary Algorithms: Empirical Results
Evolutionary Computation
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
G-Metric: an M-ary quality indicator for the evaluation of non-dominated sets
Proceedings of the 10th annual conference on Genetic and 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
Effects of 1-Greedy $\mathcal{S}$-Metric-Selection on Innumerably Large Pareto Fronts
EMO '09 Proceedings of the 5th International Conference on Evolutionary Multi-Criterion Optimization
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
An adaptive divide-and-conquer methodology for evolutionary multi-criterion optimisation
EMO'03 Proceedings of the 2nd international conference on Evolutionary multi-criterion optimization
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
SEAL'10 Proceedings of the 8th international conference on Simulated evolution and learning
Hype: An algorithm for fast hypervolume-based many-objective optimization
Evolutionary Computation
An EMO algorithm using the hypervolume measure as selection criterion
EMO'05 Proceedings of the Third international conference on Evolutionary Multi-Criterion Optimization
Properties of an adaptive archiving algorithm for storing nondominated vectors
IEEE Transactions on Evolutionary Computation
Performance assessment of multiobjective optimizers: an analysis and review
IEEE Transactions on Evolutionary Computation
IEEE Transactions on Evolutionary Computation
A review of multiobjective test problems and a scalable test problem toolkit
IEEE Transactions on Evolutionary Computation
SEAL'10 Proceedings of the 8th international conference on Simulated evolution and learning
On the properties of the R2 indicator
Proceedings of the 14th annual conference on Genetic and evolutionary computation
MOMCMC: An efficient Monte Carlo method for multi-objective sampling over real parameter space
Computers & Mathematics with Applications
On set-based local search for multiobjective combinatorial optimization
Proceedings of the 15th annual conference on Genetic and evolutionary computation
Population size matters: rigorous runtime results for maximizing the hypervolume indicator
Proceedings of the 15th annual conference on Genetic and evolutionary computation
Hi-index | 5.23 |
In recent years, indicator-based evolutionary algorithms, allowing to implicitly incorporate user preferences into the search, have become widely used in practice to solve multiobjective optimization problems. When using this type of methods, the optimization goal changes from optimizing a set of objective functions simultaneously to the single-objective optimization goal of finding a set of @m points that maximizes the underlying indicator. Understanding the difference between these two optimization goals is fundamental when applying indicator-based algorithms in practice. On the one hand, a characterization of the inherent optimization goal of different indicators allows the user to choose the indicator that meets her preferences. On the other hand, knowledge about the sets of @m points with optimal indicator values-the so-called optimal@m-distributions-can be used in performance assessment whenever the indicator is used as a performance criterion. However, theoretical studies on indicator-based optimization are sparse. One of the most popular indicators is the weighted hypervolume indicator. It allows to guide the search towards user-defined objective space regions and at the same time has the property of being a refinement of the Pareto dominance relation with the result that maximizing the indicator results in Pareto-optimal solutions only. In previous work, we theoretically investigated the unweighted hypervolume indicator in terms of a characterization of optimal @m-distributions and the influence of the hypervolume's reference point for general bi-objective optimization problems. In this paper, we generalize those results to the case of the weighted hypervolume indicator. In particular, we present general investigations for finite @m, derive a limit result for @m going to infinity in terms of a density of points and derive lower bounds (possibly infinite) for placing the reference point to guarantee the Pareto front's extreme points in an optimal @m-distribution. Furthermore, we state conditions about the slope of the front at the extremes such that there is no finite reference point that allows to include the extremes in an optimal @m-distribution-contradicting previous belief that a reference point chosen just above the nadir point or the objective space boundary is sufficient for obtaining the extremes. However, for fronts where there exists a finite reference point allowing to obtain the extremes, we show that for @m to infinity, a reference point that is slightly worse in all objectives than the nadir point is a sufficient choice. Last, we apply the theoretical results to problems of the ZDT, DTLZ, and WFG test problem suites.