Logarithmic-Time Updates in SMS-EMOA and Hypervolume-Based Archiving

Hupkens, Iris; Emmerich, Michael

doi:10.1007/978-3-319-01128-8_11

Cited by 14 publications

(6 citation statements)

References 8 publications

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“…Also it has been pointed out that the computation of a single HV contribution in dimension d has at least the complexity of the computation of the HV indicator in d − 1 Emmerich and Fonseca (2011). In addition, as shown in Hupkens and Emmerich (2013), updating all HV contributions in an archive of N nondominated solutions can be achieved with an amortised time complexity of Θ log N after adding or removing a single solution in the 2D case. For the 3D and 4D cases, algorithms are known with (amortised) runtime linear and quadratic in N , respectively Guerreiro and Fonseca (2018).…”

Section: Incremental Updatementioning

confidence: 99%

Quality Evaluation of Solution Sets in Multiobjective Optimisation

2019

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Section: Incremental Updatementioning

confidence: 99%

Quality Evaluation of Solution Sets in Multiobjective Optimisation

2019

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“…Furthermore, line 8 could simply recompute the whole contribution H(q, S∪{p}). Updating the contributions of all points p in a set X to the corresponding sets X\{p} under single-point changes to X can already be performed efficiently in 2 dimensions [11]. The following sections show how H(p, q, S) can be efficiently computed in the 2-and 3-dimensional cases.…”

Section: General Casementioning

confidence: 98%

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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“…Furthermore, the whole contribution H(q, S ∪ {p}) could be simply recomputed in line 8. Updating the contributions of all points p in a set X to the corresponding sets X\{p} under single-point changes to X can already be performed efficiently in 2 dimensions (Hupkens and Emmerich, 2013). However, in the incremental greedy algorithm, the contributions of points in X have to be updated w.r.t.…”

Section: General Casementioning

confidence: 99%

Greedy Hypervolume Subset Selection in Low Dimensions

Guerreiro

Fonseca

Paquete

2016

Evolutionary Computation

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Given a nondominated point set [Formula: see text] of size [Formula: see text] and a suitable reference point [Formula: see text], the Hypervolume Subset Selection Problem (HSSP) consists of finding a subset of size [Formula: see text] 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 [Formula: see text]. In contrast, the best upper bound available for [Formula: see text] is [Formula: see text]. Since the hypervolume indicator is a monotone submodular function, the HSSP can be approximated to a factor of [Formula: see text] using a greedy strategy. In this article, greedy [Formula: see text]-time algorithms for the HSSP in 2 and 3 dimensions are proposed, matching the complexity of current exact algorithms for the 2-dimensional case, and considerably improving upon recent complexity results for this approximation problem.

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Logarithmic-Time Updates in SMS-EMOA and Hypervolume-Based Archiving

Cited by 14 publications

References 8 publications

Quality Evaluation of Solution Sets in Multiobjective Optimisation

Quality Evaluation of Solution Sets in Multiobjective Optimisation

Greedy Hypervolume Subset Selection in the Three-Objective Case

Greedy Hypervolume Subset Selection in Low Dimensions

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