## Using Unconstrained Elite Archives for Multi-Objective Optimisation (2001)

Venue: | IEEE Transactions on Evolutionary Computation |

Citations: | 38 - 12 self |

### BibTeX

@ARTICLE{Fieldsend01usingunconstrained,

author = {Jonathan E. Fieldsend and Richard M. Everson and Sameer Singh},

title = {Using Unconstrained Elite Archives for Multi-Objective Optimisation},

journal = {IEEE Transactions on Evolutionary Computation},

year = {2001},

volume = {7},

pages = {305--323}

}

### Years of Citing Articles

### OpenURL

### Abstract

Multi-Objective Evolutionary Algorithms (MOEAs) have been the subject of numer- ous studies over the past 20 years. Recent work has highlighted the use of an active archive of elite, non-dominated solutions to improve the optimisation speed of these algorithms.

### Citations

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Citation Context ...g frontal individuals that lie to the south-west' or qeft and below' y. More formally, the set of dominating individuals is: {z E F : zi _ Yi for all 1sj _ D) (8) It would be possible to use kd-trees =-=[26, 25]-=- or range trees [25, 27], but these are both suited to querying F for elements which lie in bounded (hyper-) rectangles. Priority trees, developed by McCreight [28], are suited to rectangular queries ... |

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Citation Context ...al objectives to be optimised. In addition, EAs, which maintain a population of solutions, are able to explore several parts of the Pareto front simultaneously. Most recent investigations in the area =-=[1, 7, 10, 11, 12]-=- focus on a MOEA's ability to produce an accurate estimate of the Pareto front. Zitzler et al. [9] present a comparative study, on six test functions introduced by Deb [13], of a number of the most wi... |

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Citation Context ...of objectives in an aircraft simulator [5]. However, it is the work of Schaier in 1985 [6], which recognised the need to return a set of solutions, that has been widely quoted as the first MOEA study =-=[7, 8, 9]-=-. The use of Evolutionary Algorithms (EAs) as the tool of choice is due to such problems being typically complex, with both a large number of parameters to be adjusted, and several objectives to be op... |

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Citation Context ...al objectives to be optimised. In addition, EAs, which maintain a population of solutions, are able to explore several parts of the Pareto front simultaneously. Most recent investigations in the area =-=[1, 7, 10, 11, 12]-=- focus on a MOEA's ability to produce an accurate estimate of the Pareto front. Zitzler et al. [9] present a comparative study, on six test functions introduced by Deb [13], of a number of the most wi... |

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Citation Context ...of objectives in an aircraft simulator [5]. However, it is the work of Schaier in 1985 [6], which recognised the need to return a set of solutions, that has been widely quoted as the first MOEA study =-=[7, 8, 9]-=-. The use of Evolutionary Algorithms (EAs) as the tool of choice is due to such problems being typically complex, with both a large number of parameters to be adjusted, and several objectives to be op... |

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Citation Context ...me now: Beale and Cook in 1978 used a random search technique in an attempt to simultaneously miniraise a number of objectives in an aircraft simulator [5]. However, it is the work of Schaier in 1985 =-=[6]-=-, which recognised the need to return a set of solutions, that has been widely quoted as the first MOEA study [7, 8, 9]. The use of Evolutionary Algorithms (EAs) as the tool of choice is due to such p... |

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Citation Context ...a non-dominated set (an estimate of the Pareto front) if no member of the set is dominated by any other member: wiwj Vi,j=i,...,M (7) 3 THE UNIFIED MODEL As evinced by a number of comparative studies =-=[9, 14, 15]-=- the SPEA provides an effective methodology for multi-objective optimisation problems, as does the PAES [16, 17]. Both algorithms can be seen as variants of the Unified Model for Multi-objective Evolu... |

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Citation Context ...of objectives in an aircraft simulator [5]. However, it is the work of Schaier in 1985 [6], which recognised the need to return a set of solutions, that has been widely quoted as the first MOEA study =-=[7, 8, 9]-=-. The use of Evolutionary Algorithms (EAs) as the tool of choice is due to such problems being typically complex, with both a large number of parameters to be adjusted, and several objectives to be op... |

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Citation Context ...al objectives to be optimised. In addition, EAs, which maintain a population of solutions, are able to explore several parts of the Pareto front simultaneously. Most recent investigations in the area =-=[1, 7, 10, 11, 12]-=- focus on a MOEA's ability to produce an accurate estimate of the Pareto front. Zitzler et al. [9] present a comparative study, on six test functions introduced by Deb [13], of a number of the most wi... |

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Citation Context ...other MOEAs on a 0/1 knapsack problem. Another MOEA which has demonstrated significant results is the Evolutionary Strategy (ES) based Pareto Archived Evolutionary Strategy (PAES) of Knowles and Come =-=[16, 17]-=-. Both these MOEAs incorporate elitism and recent work by Laumanns et al. [18] provides a unified model for MOEAs with elitism (called UMMEA). The elitism within UMMEA is achieved by using an archive ... |

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Citation Context ...d Cook [5] include a fitness-based stopping criterion in their study, in which the algorithm is terminated if the fittest individual has remained unchanged for 1000 consecutive generations. As Coello =-=[10]-=- points out, MOEAs since Schaffer [6], which carry a set of non-dominated individuals, are usually terminated after a fixed number of generations, or the population is monitored at intervals and a dec... |

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Citation Context ...uld be possible to use kd-trees [26, 25] or range trees [25, 27], but these are both suited to querying F for elements which lie in bounded (hyper-) rectangles. Priority trees, developed by McCreight =-=[28]-=-, are suited to rectangular queries in which the rectangle is unbounded on a single side. Sun and Steuer [29] describe an alternative data structure adapted for answering queries about domination and ... |

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Citation Context ...s in the area [1, 7, 10, 11, 12] focus on a MOEA's ability to produce an accurate estimate of the Pareto front. Zitzler et al. [9] present a comparative study, on six test functions introduced by Deb =-=[13]-=-, of a number of the most widely used MOEAs, including Fonseca and Fieming's multiobjective EA (FFGA) [1], the Niched Pareto Genetic Algorithm (NPGA) [11], Hajela and Lin's weighted-sum approach (HLGA... |

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Citation Context ...other MOEAs on a 0/1 knapsack problem. Another MOEA which has demonstrated significant results is the Evolutionary Strategy (ES) based Pareto Archived Evolutionary Strategy (PAES) of Knowles and Come =-=[16, 17]-=-. Both these MOEAs incorporate elitism and recent work by Laumanns et al. [18] provides a unified model for MOEAs with elitism (called UMMEA). The elitism within UMMEA is achieved by using an archive ... |

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Citation Context ... two objective problems. 26 For the purposes of this study, where the focus is on maintaining large archives, the ZDT functions unfortunately are arguably of limited use (note however that Deb et al. =-=[32]-=- have recently introduced additional higher dimension test functions). The first three ZDT functions are still employed as they are well known to the community at large. In addition three new test fun... |

79 | On metrics for comparing nondominated sets
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Citation Context ...ed, it is unlikely that any one measure will be sufficient to encompass all desired information when evaluating the output of an MOEA (a comparison of a number of widely used measures can be found in =-=[33]-=-). In this study two performance measures are used to encapsulate different properties of competing non-dominated sets. First, an alternative to the popular C metric [9, 14, 15] is discussed. The C me... |

41 |
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Citation Context ...aining a large archive. It is shown in this paper that a consequence of restricting the number of solutions in the elite front can be shrinking [19] and oscillating/retreating estimated Pareto fronts =-=[20, 21, 22]-=-. These problems also occur in SPEA, PAES and other existing MOEAs which use a truncated elite archive. A remedy to this situation is simply to retain all the non-dominated solutions found (as an acti... |

31 |
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Citation Context ...t). 2 In Ill for example, multi-objective optimisation is applied to four performance measures of a gas turbine, in [2] different loads in trusses are the competing objectives to be miniraised and in =-=[3]-=- different properties of a pressurised water reactor load pattern are optimised. The curve (for two objectives) or surface (more than two objectives) that describes the optimal trade-off possibilities... |

31 | 2002) “Running Performance Metrics for Evolutionary Multi-Objective Optimization”, Indian Institute of Technology, Kanpur: Kanpur Genetic Algorithms Laboratory (KanGAL
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Citation Context ...ins isolated points in objective space. 36 spurious early termination. (An alternative discussion on stopping techniques for MOEAs, through the use of multiple run-time error metrics, can be found in =-=[36]-=-). 8 DISCUSSION New data structures and methods - PQRS and dominated trees - have been introduced to facilitate faster performance when maintaining an unconstrained archive of non-dominated solutions.... |

30 | L.: A unified model for multiobjective evolutionary algorithms with elitism
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Citation Context ...cant results is the Evolutionary Strategy (ES) based Pareto Archived Evolutionary Strategy (PAES) of Knowles and Come [16, 17]. Both these MOEAs incorporate elitism and recent work by Laumanns et al. =-=[18]-=- provides a unified model for MOEAs with elitism (called UMMEA). The elitism within UMMEA is achieved by using an archive set of non-dominated solutions in addition to the usual GA population (or as a... |

19 |
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Citation Context ...ing the GA. This is a reflection of the size of the archive set - see figures 11 and 12. Table 2 shows the generations at which one algorithm tested using the nonparametric Wilcoxon Signed Ranks Test =-=[35]-=- at the 5% level (2.5% in each tail) 5 was found to be significantly faster than another. As can be seen, both the unconstrained algorithms were initially faster than the clustered approach, however, ... |

17 |
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Citation Context ...2) case. Dominated trees also have applications to general sets (that is, sets whose elements are not mutually non-dominating), such as answering queries about enclosing rectangles; see, for example, =-=[28, 31]-=-. 5.3.1 Construction Algorithm 2 Construction of a dominated tree Inputs: F= M {Ym}m= 1: T:= 2: c = 3: for i := 1,...,L 4: for d := 1,...,D 5: y := maxymcF(Ym,d ) 6: Ci,d : Yd 7: if IF[s1 8: F:=Fy 9: ... |

14 | On the convergence and diversity-preservation properties of multi-objective evolutionary algorithms
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(Show Context)
Citation Context ...aining a large archive. It is shown in this paper that a consequence of restricting the number of solutions in the elite front can be shrinking [19] and oscillating/retreating estimated Pareto fronts =-=[20, 21, 22]-=-. These problems also occur in SPEA, PAES and other existing MOEAs which use a truncated elite archive. A remedy to this situation is simply to retain all the non-dominated solutions found (as an acti... |

10 |
d’économie politique
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(Show Context)
Citation Context ...tor load pattern are optimised. The curve (for two objectives) or surface (more than two objectives) that describes the optimal trade-off possibilities between objectives is known as the Pareto front =-=[4]-=-. A feasible solution lying on the Pareto front cannot improve any objective without degrading at least one of the others, and, given the constraints of the model, no solutions exist beyond the true P... |

9 | Comparison of data structures for storing Pareto sets
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(Show Context)
Citation Context ...ed on a single side. Sun and Steuer [29] describe an alternative data structure adapted for answering queries about domination and non-domination; which has been extended recently by Mostaghim et al. =-=[30]-=-. The dominates' relation imposes a partial order on individuals. However, since the elements of F are mutually non-dominating, this relation cannot be used directly to construct, for example, a binar... |

8 |
multi-objective genetic algorithms
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(Show Context)
Citation Context ...ropping the trttcate 0 operation. This set of all currently non-dominated individuals is referred to as the jotl set F. This approach has previously been adopted by, for example, Murata and Ishibuchi =-=[24]-=- and Parks and Miller [3], and it should be noted that, while Laumanns et al [18] recommend the truncation of the external set, one of their numerical studies retains all non-dominated individuals. Ho... |

8 |
Lebesgue Integration on Euclidean
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(Show Context)
Citation Context ...: z-(hy(a) forsomeaA / (32) be the set of points in the hypercube defined by Y which are dominated by the normalised A. Then 12 (A, B) is defined as (33) where ),(A) denotes the Lebesgue measure of A =-=[34]-=-. Despite this rather cumbersome description, 12 (A, B) and 12 (B, A) are easily calculated by Monte Carlo sampling of HAUB and counting the fraction of samples that are dominated exclusively by A or ... |

6 | Pareto multi-objective non-linear regression modelling to aid CAPM analogous forecasting
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(Show Context)
Citation Context ...facilitate multi-objective particle swarm optimisation [37]. The data structures and methods introduced in this study are currently being applied to the evolution of neural network forecasting models =-=[38]-=-, information retrieval and the optimisation of safety critical systems. ACKNOWLEDGEMENTS Jonathan Fieldsend gratefully acknowledges support from Invensys Climate Controls Europe and an Exeter Univers... |

5 |
Global Multiobjective Optimization Using Evolutionary Algorithms
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(Show Context)
Citation Context ...s the front advances in the MOEA search stage but retreats during truncation. This possibility was first noted by Hanne [20] in different situation. In Hanne's MOEA (an applied example of which is in =-=[23]-=-) a (;k-t-/a)-ES scheme was used, with a child replacing the parent if the parent does not dominate the child. As such the population that Hanne used was not a Pareto archive as it was not a non-domin... |

4 | S.Singh. Full Elite Sets for Multi-Objective Optimisation
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(Show Context)
Citation Context ...aining a large archive. It is shown in this paper that a consequence of restricting the number of solutions in the elite front can be shrinking [19] and oscillating/retreating estimated Pareto fronts =-=[20, 21, 22]-=-. These problems also occur in SPEA, PAES and other existing MOEAs which use a truncated elite archive. A remedy to this situation is simply to retain all the non-dominated solutions found (as an acti... |

3 |
Data structures for range searching. Computing Surveys
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(Show Context)
Citation Context ...at lie to the south-west' or qeft and below' y. More formally, the set of dominating individuals is: {z E F : zi _ Yi for all 1sj _ D) (8) It would be possible to use kd-trees [26, 25] or range trees =-=[25, 27]-=-, but these are both suited to querying F for elements which lie in bounded (hyper-) rectangles. Priority trees, developed by McCreight [28], are suited to rectangular queries in which the rectangle i... |

2 |
Optimal Digital Simulation of Aircraft via Random Search Techniques
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(Show Context)
Citation Context ...have been in use for a considerable length of time now: Beale and Cook in 1978 used a random search technique in an attempt to simultaneously miniraise a number of objectives in an aircraft simulator =-=[5]-=-. However, it is the work of Schaier in 1985 [6], which recognised the need to return a set of solutions, that has been widely quoted as the first MOEA study [7, 8, 9]. The use of Evolutionary Algorit... |

1 |
InterQuadi: An ineteractive quad tree based procedure for solving the discrete alternative multiple criteria problem
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(Show Context)
Citation Context ... elements which lie in bounded (hyper-) rectangles. Priority trees, developed by McCreight [28], are suited to rectangular queries in which the rectangle is unbounded on a single side. Sun and Steuer =-=[29]-=- describe an alternative data structure adapted for answering queries about domination and non-domination; which has been extended recently by Mostaghim et al. [30]. The dominates' relation imposes a ... |