Hypervolume Subset Selection in Two Dimensions: Formulations and Algorithms

Kühn, Tobias; Fonseca, Carlos M.; Paquete, Luís; Ruzika, Stefan; Duarte, Miguel M.; Figueira, José Rui

doi:10.1162/evco_a_00157

Cited by 62 publications

(34 citation statements)

References 26 publications

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“…Extensions towards other quality measures that can be efficiently computed on-the-fly are planned. It is certainly interesting to additionally use the highest indicator value over all subsets of the archive with exactly μ solutions as a performance criterion as suggested in [2,13]. …”

Section: Discussionmentioning

confidence: 99%

“…For these optimal sets of μ solutions with respect to a quality indicator, the term optimal μ-distribution has been introduced [1]. If, on the other hand, an unbounded set with maximal quality indicator is sought, a good idea is to maintain an external archive of all non-dominated solutions found so far which, recently, even has been argued for when the the optimal μ-distribution is sought and can be extracted from the archive efficiently [2,13].…”

Section: Transferring Single-objective Benchmarking Concepts To the Mmentioning

confidence: 99%

“…5.2). Note finally, that in the future, we plan to include also the extracted best subset of the archive of a specific size as performance measure, similar to [2,13].…”

Section: Generalizing the Bbob Testbed To Multiobjective Optimizationmentioning

confidence: 99%

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Benchmarking Numerical Multiobjective Optimizers Revisited

Brockhoff

Tran

Hansen

2015

Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation

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International audienceAlgorithm benchmarking plays a vital role in designing new optimization algorithms and in recommending efficient and robust algorithms for practical purposes. So far, two main approaches have been used to compare algorithms in the evolutionary multiobjective optimization (EMO) field: (i) displaying empirical attainment functions and (ii) reporting statistics on quality indicator values. Most of the time, EMO benchmarking studies compare algorithms for fixed and often arbitrary budgets of function evaluations although the algorithms are anytime optimizers. Instead, we propose to transfer and adapt standard benchmarking techniques from the single-objective optimization and classical derivative-free optimization community to the field of EMO. Reporting target-based runlengths allows to compare algorithms with varying numbers of function evaluations quantitatively. Displaying data profiles can aggregate performance information over different test functions, problem difficulties, and quality indicators. We apply this approach to compare three common algorithms on a new test function suite derived from the well-known single-objective BBOB functions. The focus thereby lies less on gaining insights into the algorithms but more on showcasing the concepts and on what can be gained over current benchmarking approaches

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Section: Discussionmentioning

confidence: 99%

Section: Transferring Single-objective Benchmarking Concepts To the Mmentioning

confidence: 99%

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Benchmarking Numerical Multiobjective Optimizers Revisited

Brockhoff

Tran

Hansen

2015

Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation

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“…This is known as the Hypervolume Subset Selection Problem (HSSP) [2], and algorithms to solve it exactly are available only for 2 dimensions, with O(n(k + log n))-time complexity [6,13]. For d > 2, the equivalent problem of determining a subset of n − k points that contribute the least hypervolume to the original set can be solved in O(n d 2 log n + n n−k ) [5].…”

Section: Introductionmentioning

confidence: 99%

Greedy Hypervolume Subset Selection in the Three-Objective Case

Guerreiro

Fonseca

Paquete

2015

Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation

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Given a non-dominated point set X ⊂ R d of size n and a suitable reference point r ∈ R d , the Hypervolume Subset Selection Problem (HSSP) consists of finding a subset of size k ≤ n that maximizes the hypervolume indicator. It arises in connection with multiobjective selection and archiving strategies, as well as Pareto-front approximation postprocessing for visualization and/or interaction with a decision maker. Efficient algorithms to solve the HSSP are available only for the 2-dimensional case, achieving a time complexity of O(n(k + log n)). In contrast, the best upper bound available for d > 2 is O(n d 2 log n + n n−k ). Since the hypervolume indicator is a monotone submodular function, the HSSP can be approximated to a factor of (1 − 1/e) using a greedy strategy. Such a greedy algorithm for the 3-dimensional HSSP is proposed in this paper. The time complexity of the algorithm is shown to be O(n 2 ), which considerably improves upon recent complexity results for this approximation problem.

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“…For 2 dimensions there are algorithms which solve HYPSSP in time O(n 3 ) [4], O(kn 2 ) [3] and O(n 2 ) [16]. The only known lower bound is the trivial Ω(n).…”

Section: Introductionmentioning

confidence: 99%

Two-dimensional subset selection for hypervolume and epsilon-indicator

Bringmann

Friedrich

Klitzke

2014

Proceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation

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The goal of bi-objective optimization is to find a small set of good compromise solutions. A common problem for bi-objective evolutionary algorithms is the following subset selection problem (SSP): Given n solutions P ⊂ R 2 in the objective space, select k solutions P * from P that optimize an indicator function. In the hypervolume SSP we want to select k points P * that maximize the hypervolume indicator Ihyp(P * , r) for some reference point r ∈ R 2 . Similarly, the ε-indicator SSP aims at selecting k points P * that minimize the ε-indicator Ieps(P * , R) for some reference set R ⊂ R 2 of size m (which can be R = P ).We first present a new algorithm for the hypervolume SSP with runtime O(n (k + log n)). Our second main result is a new algorithm for the ε-indicator SSP with runtime O(n log n + m log m).Both results improve the current state of the art runtimes by a factor of (nearly) n and make the problems tractable for new applications. Preliminary experiments confirm that the theoretical results translate into substantial empirical runtime improvements.

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Hypervolume Subset Selection in Two Dimensions: Formulations and Algorithms

Cited by 62 publications

References 26 publications

Benchmarking Numerical Multiobjective Optimizers Revisited

Benchmarking Numerical Multiobjective Optimizers Revisited

Greedy Hypervolume Subset Selection in the Three-Objective Case

Two-dimensional subset selection for hypervolume and epsilon-indicator

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