Introduction to numerical analysis: 2nd edition
Introduction to numerical analysis: 2nd edition
A new approach to the dynamic maintenance of maximal points in a plane
Discrete & Computational Geometry
Modern heuristic techniques for combinatorial problems
Modern heuristic techniques for combinatorial problems
Dynamic Maintenance of Maxima of 2-d Point Sets
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
On the Performance Assessment and Comparison of Stochastic Multiobjective Optimizers
PPSN IV Proceedings of the 4th International Conference on Parallel Problem Solving from Nature
Archiving With Guaranteed Convergence And Diversity In Multi-objective Optimization
GECCO '02 Proceedings of the Genetic and Evolutionary Computation Conference
Multiobjective Evolutionary Algorithms: Analyzing the State-of-the-Art
Evolutionary Computation
Approximating the Nondominated Front Using the Pareto Archived Evolution Strategy
Evolutionary Computation
GECCO '05 Proceedings of the 7th annual conference on Genetic and evolutionary computation
Methods for multi-dimensional robustness optimization in complex embedded systems
EMSOFT '07 Proceedings of the 7th ACM & IEEE international conference on Embedded software
Quality Assessment of Pareto Set Approximations
Multiobjective Optimization
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
EvoBIO '09 Proceedings of the 7th European Conference on Evolutionary Computation, Machine Learning and Data Mining in Bioinformatics
Spread Assessment for Evolutionary Multi-Objective Optimization
EMO '09 Proceedings of the 5th International Conference on Evolutionary Multi-Criterion Optimization
Articulating user preferences in many-objective problems by sampling the weighted hypervolume
Proceedings of the 11th Annual conference on Genetic and evolutionary computation
Investigating and exploiting the bias of the weighted hypervolume to articulate user preferences
Proceedings of the 11th Annual conference on Genetic and evolutionary computation
Convergence acceleration operator for multiobjective optimization
IEEE Transactions on Evolutionary Computation
S-metric calculation by considering dominated hypervolume as klee's measure problem
Evolutionary Computation
On the complexity of computing the hypervolume indicator
IEEE Transactions on Evolutionary Computation
Correction to "a fast incremental hypervolume algorithm"
IEEE Transactions on Evolutionary Computation
EMO'07 Proceedings of the 4th international conference on Evolutionary multi-criterion optimization
HM'07 Proceedings of the 4th international conference on Hybrid metaheuristics
Computers and Operations Research
PPSN'10 Proceedings of the 11th international conference on Parallel problem solving from nature: Part I
A novel smart multi-objective particle swarm optimisation using decomposition
PPSN'10 Proceedings of the 11th international conference on Parallel problem solving from nature: Part II
Expert Systems with Applications: An International Journal
SEAL'10 Proceedings of the 8th international conference on Simulated evolution and learning
Ectropy of diversity measures for populations in Euclidean space
Information Sciences: an International Journal
Hype: An algorithm for fast hypervolume-based many-objective optimization
Evolutionary Computation
On sequential online archiving of objective vectors
EMO'11 Proceedings of the 6th international conference on Evolutionary multi-criterion optimization
EMO'11 Proceedings of the 6th international conference on Evolutionary multi-criterion optimization
Proceedings of the 13th annual conference companion on Genetic and evolutionary computation
Achieving balance between proximity and diversity in multi-objective evolutionary algorithm
Information Sciences: an International Journal
An EMO algorithm using the hypervolume measure as selection criterion
EMO'05 Proceedings of the Third international conference on Evolutionary Multi-Criterion Optimization
A new analysis of the lebmeasure algorithm for calculating hypervolume
EMO'05 Proceedings of the Third international conference on Evolutionary Multi-Criterion Optimization
EMO'05 Proceedings of the Third international conference on Evolutionary Multi-Criterion Optimization
A two-level evolutionary approach to multi-criterion optimization of water supply systems
EMO'05 Proceedings of the Third international conference on Evolutionary Multi-Criterion Optimization
Hypervolume-based multiobjective optimization: Theoretical foundations and practical implications
Theoretical Computer Science
On Klee's measure problem for grounded boxes
Proceedings of the twenty-eighth annual symposium on Computational geometry
A new multi-objective evolutionary algorithm based on a performance assessment indicator
Proceedings of the 14th annual conference on Genetic and evolutionary computation
A multi-objective artificial immune system based on hypervolume
ICARIS'12 Proceedings of the 11th international conference on Artificial Immune Systems
Clustering-Based multi-objective immune optimization evolutionary algorithm
ICARIS'12 Proceedings of the 11th international conference on Artificial Immune Systems
Advances in evolutionary multi-objective optimization
SSBSE'12 Proceedings of the 4th international conference on Search Based Software Engineering
On set-based local search for multiobjective combinatorial optimization
Proceedings of the 15th annual conference on Genetic and evolutionary computation
Engineering Applications of Artificial Intelligence
Annals of Mathematics and Artificial Intelligence
Computational Statistics & Data Analysis
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This article describes a set function that maps a set of Pareto optimal points to a scalar. A theorem1 is presented that shows that the maximization of this scalar value constitutes the necessary and sufficient condition for the function's arguments to be maximally diverse Pareto optimal solutions of a discrete, multi-objective, optimization problem. This scalar quantity, a hypervolume based on a Lebesgue measure, is therefore the best metric to assess the quality of multi-objective optimization algorithms. Moreover, it can be used as the objective function in simulated annealing (SA) to induce convergence in probability to the Pareto optima. An efficient, polynomial-time algorithm for calculating this scalar and an analysis of its complexity is also presented.