20:
166:. Nonetheless, the advantage of EDAs is also that these algorithms provide an optimization practitioner with a series of probabilistic models that reveal a lot of information about the problem being solved. This information can in turn be used to design problem-specific neighborhood operators for local search, to bias future runs of EDAs on a similar problem, or to create an efficient computational model of the problem.
4180:
equivalence (BDe)). The scoring metric evaluates the network structure according to its accuracy in modeling the selected population. From the built network, BOA samples new promising solutions as follows: (1) it computes the ancestral ordering for each variable, each node being preceded by its parents; (2) each variable is sampled conditionally to its parents. Given such scenario, every BOA step can be defined as
169:
For example, if the population is represented by bit strings of length 4, the EDA can represent the population of promising solution using a single vector of four probabilities (p1, p2, p3, p4) where each component of p defines the probability of that position being a 1. Using this probability vector
78:
methods that guide the search for the optimum by building and sampling explicit probabilistic models of promising candidate solutions. Optimization is viewed as a series of incremental updates of a probabilistic model, starting with the model encoding an uninformative prior over admissible solutions
4179:
The
Bayesian network structure, on the other hand, must be built iteratively (linkage-learning). It starts with a network without edges and, at each step, adds the edge which better improves some scoring metric (e.g. Bayesian information criterion (BIC) or Bayesian-Dirichlet metric with likelihood
3849:
The linkage-learning in ECGA works as follows: (1) Insert each variable in a cluster, (2) compute CCC = MC + CPC of the current linkage sets, (3) verify the increase on CCC provided by joining pairs of clusters, (4) effectively joins those clusters with highest CCC improvement. This procedure is
3540:
The ECGA popularized the term "linkage-learning" as denoting procedures that identify linkage sets. Its linkage-learning procedure relies on two measures: (1) the Model
Complexity (MC) and (2) the Compressed Population Complexity (CPC). The MC quantifies the model representation size in terms of
2690:
The BMDA factorizes the joint probability distribution in bivariate distributions. First, a randomly chosen variable is added as a node in a graph, the most dependent variable to one of those in the graph is chosen among those not yet in the graph, this procedure is repeated until no remaining
1951:
Although univariate models can be computed efficiently, in many cases they are not representative enough to provide better performance than GAs. In order to overcome such a drawback, the use of bivariate factorizations was proposed in the EDA community, in which dependencies between pairs of
5352:
The LTGA does not implement typical selection operators, instead, selection is performed during recombination. Similar ideas have been usually applied into local-search heuristics and, in this sense, the LTGA can be seen as an hybrid method. In summary, one step of the LTGA is defined as
2592:
New solutions are sampled from the leftmost to the rightmost variable, the first is generated independently and the others according to conditional probabilities. Since the estimated distribution must be recomputed each generation, MIMIC uses concrete populations in the following way
2979:
1195:
3305:
The ECGA was one of the first EDA to employ multivariate factorizations, in which high-order dependencies among decision variables can be modeled. Its approach factorizes the joint probability distribution in the product of multivariate marginal distributions. Assume
161:
Using explicit probabilistic models in optimization allowed EDAs to feasibly solve optimization problems that were notoriously difficult for most conventional evolutionary algorithms and traditional optimization techniques, such as problems with high levels of
3971:
The BOA uses
Bayesian networks to model and sample promising solutions. Bayesian networks are directed acyclic graphs, with nodes representing variables and edges representing conditional probabilities between pair of variables. The value of a variable
3276:
The learning of PGMs encoding multivariate distributions is a computationally expensive task, therefore, it is usual for EDAs to estimate multivariate statistics from bivariate statistics. Such relaxation allows PGM to be built in polynomial time in
4494:
849:
2588:
1759:
2178:
1942:
3653:
522:
941:
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4175:
2681:
5450:
4046:. BOA builds a PGM encoding a factorized joint distribution, in which the parameters of the network, i.e. the conditional probabilities, are estimated from the selected population using the maximum likelihood estimator.
3071:
1369:
3962:
4265:
3272:
3845:
3369:
2780:
6638:
Yu, Tian-Li; Goldberg, David E.; Yassine, Ali; Chen, Ying-Ping (2003), "Genetic
Algorithm Design Inspired by Organizational Theory: Pilot Study of a Dependency Structure Matrix Driven Genetic Algorithm",
3157:
1004:
386:
6426:
Iacca, Giovanni; Mallipeddi, Rammohan; Mininno, Ernesto; Neri, Ferrante; Suganthan, Pannuthurai
Nagaratnam (2011). "Super-fit and population size reduction in compact Differential Evolution".
3080:
The next stage of EDAs development was the use of multivariate factorizations. In this case, the joint probability distribution is usually factorized in a number of components of limited size
2311:
2186:(graphs), in which edges denote statistical dependencies (or conditional probabilities) and vertices denote variables. To learn the structure of a PGM from data linkage-learning is employed.
2748:
the non-root variables, BMDA estimates a factorized distribution in which the root variables can be sampled independently, whereas all the others must be conditioned to the parent variable
5022:
6234:
Iacca, Giovanni; Caraffini, Fabio; Neri, Ferrante (2012). "Compact
Differential Evolution Light: High Performance Despite Limited Memory Requirement and Modest Computational Overhead".
2385:
1567:
3402:
5220:
4577:
1482:
4323:
997:
707:
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1235:
102:, or another model class. Similarly as other evolutionary algorithms, EDAs can be used to solve optimization problems defined over a number of representations from vectors to
4634:
3875:
2392:
700:
568:
6385:
Iacca, Giovanni; Mallipeddi, Rammohan; Mininno, Ernesto; Neri, Ferrante; Suganthan, Pannuthurai
Nagaratnam (2011). "Global supervision for compact Differential Evolution".
6309:
Mallipeddi, Rammohan; Iacca, Giovanni; Suganthan, Ponnuthurai
Nagaratnam; Neri, Ferrante; Mininno, Ernesto (2011). "Ensemble strategies in Compact Differential Evolution".
4951:
4772:
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2004:
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1511:
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205:
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4292:
4017:
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1607:
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1396:
406:
245:
225:
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4274:
The LTGA differs from most EDA in the sense it does not explicitly model a probability distribution but only a linkage model, called linkage-tree. A linkage
6350:
Neri, Ferrante; Iacca, Giovanni; Mininno, Ernesto (2011). "Disturbed
Exploitation compact Differential Evolution for limited memory optimization problems".
413:
86:. The main difference between EDAs and most conventional evolutionary algorithms is that evolutionary algorithms generate new candidate solutions using an
858:
3445:
4051:
2598:
5358:
2988:
1289:
6002:
WOLPERT, DAVID H.; STRAUSS, CHARLIE E. M.; RAJNARAYAN, DEV (December 2006). "Advances in
Distributed Optimization Using Probability Collectives".
3882:
4185:
5844:
Pelikan, Martin; Goldberg, David E.; Cantu-Paz, Erick (1 January 1999). "BOA: The Bayesian Optimization Algorithm". Morgan Kaufmann: 525–532.
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5811:
5600:
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946:
953:
The PBIL, represents the population implicitly by its model, from which it samples new solutions and updates the model. At each generation,
3780:
2974:{\displaystyle p_{t+1}(X_{1},\dots ,X_{N})=\prod _{X_{i}\in \Upsilon _{t}}p_{t}(X_{i})\cdot \prod _{X_{i}\in I_{t}}p_{t}(X_{i}|\pi _{i}).}
6597:
Tamayo-Vera, Dania; Bolufe-Rohler, Antonio; Chen, Stephen (2016). "Estimation multivariate normal algorithm with thresheld convergence".
3657:
The CPC, on the other hand, quantifies the data compression in terms of entropy of the marginal distribution over all partitions, where
3309:
4294:
is a set of linkage sets with no probability distribution associated, therefore, there is no way to sample new solutions directly from
6167:
6039:
Pelikan, Martin; Goldberg, David E.; Lobo, Fernando G. (2002). "A Survey of Optimization by Building and Using Probabilistic Models".
5625:
5698:"Population-Based Incremental Learning: A Method for Integrating Genetic Search Based Function Optimization and Competitive Learning"
178:
This section describes the models built by some well known EDAs of different levels of complexity. It is always assumed a population
6728:
6698:
6656:
6614:
6527:
6443:
6402:
6326:
6294:
5759:
Bonet, Jeremy S. De; Isbell, Charles L.; Viola, Paul (1 January 1996). "MIMIC: Finding Optima by Estimating Probability Densities".
5551:
388:. Therefore, univariate EDAs rely only on univariate statistics and multivariate distributions must be factorized as the product of
4609:
to guide an "optimal mixing" procedure which resembles a recombination operator but only accepts improving moves. We denote it as
6733:
3877:. The ECGA works with concrete populations, therefore, using the factorized distribution modeled by ECGA, it can be described as
3083:
99:
2198:
in a chain-like model representing successive dependencies between variables. It finds a permutation of the decision variables,
1190:{\displaystyle p_{t+1}(X_{i})=(1-\gamma )p_{t}(X_{i})+(\gamma /\lambda )\sum _{x\in S(P(t))}x_{i},~\forall i\in 1,2,\dots ,N,}
2195:
6108:
Rudlof, Stephan; Köppen, Mario (1997). "Stochastic Hill Climbing with Learning by Vectors of Normal Distributions": 60––70.
6066:
Rudlof, Stephan; Köppen, Mario (1997). "Stochastic Hill Climbing with Learning by Vectors of Normal Distributions": 60–70.
298:
5318:
6277:
Iacca, Giovanni; Neri, Ferrante; Mininno, Ernesto (2011), "Opposition-Based Learning in Compact Differential Evolution",
2314:
2233:
106:
style S expressions, and the quality of candidate solutions is often evaluated using one or more objective functions.
4959:
2320:
1516:
6191:
Mininno, Ernesto; Neri, Ferrante; Cupertino, Francesco; Naso, David (2011). "Compact Differential Evolution".
3374:
5917:
Proceedings of the 17th AAAI Conference on Late-Breaking Developments in the Field of Artificial Intelligence
5169:
4549:
4499:
83:
75:
6673:
Optimization by Pairwise Linkage Detection, Incremental Linkage Set, and Restricted / Back Mixing: DSMGA-II
1427:
6109:
6067:
6011:
5845:
5789:
5764:
4489:{\displaystyle T_{\text{LT}}=\{\{x_{1}\},\{x_{2}\},\{x_{3}\},\{x_{4}\},\{x_{1},x_{2}\},\{x_{3},x_{4}\}\}.}
844:{\displaystyle p_{t+1}(X_{i})={\dfrac {1}{\lambda }}\sum _{x\in S(P(t))}x_{i},~\forall i\in 1,2,\dots ,N.}
1378:
The CGA, also relies on the implicit populations defined by univariate distributions. At each generation
6123:
6081:
5859:
5705:
2583:{\displaystyle p_{t+1}(X_{1},\dots ,X_{N})=p_{t}(X_{r(N)})\prod _{i=1}^{N-1}p_{t}(X_{r(i)}|X_{r(i+1)}).}
5784:
Pelikan, Martin; Muehlenbein, Heinz (1 January 1999). "The Bivariate Marginal Distribution Algorithm".
5618:
Scalable optimization via probabilistic modeling : from algorithms to applications; with 26 tables
976:
5888:
Hierarchical Bayesian optimization algorithm : toward a new generation of evolutionary algorithms
5538:, Studies in Fuzziness and Soft Computing, vol. 170, Springer Berlin Heidelberg, pp. 13–30,
4546:
are merged, this procedure repeats until only one cluster remains, each subtree is stored as a subset
1766:
1202:
19:
6502:
Iacca, Giovanni; Neri, Ferrante; Mininno, Ernesto (2012), "Compact Bacterial Foraging Optimization",
5514:
4612:
3853:
6016:
669:
537:
6114:
6072:
5850:
5794:
5769:
4929:
4750:
4585:
3407:
2697:
5830:
Learning Gene Linkage to Efficiently Solve Problems of Bounded Difficulty Using Genetic Algorithms
5227:
2201:
2009:
1244:
6704:
6676:
6620:
6579:
6467:
Neri, Ferrante; Mininno, Ernesto; Iacca, Giovanni (2013). "Compact Particle Swarm Optimization".
6449:
6408:
6332:
6259:
6216:
6173:
5951:
5678:
1808:
1754:{\displaystyle p_{t+1}(X_{i})=p_{t}(X_{i})+\gamma (u_{i}-v_{i}),\quad \forall i\in 1,2,\dots ,N,}
5026:
6694:
6652:
6610:
6571:
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5662:
5621:
5596:
5571:
5547:
5060:
4639:
4022:
3730:
3660:
2751:
2173:{\displaystyle D_{\text{Bivariate}}:=p(X_{1},\dots ,X_{N})=\prod _{i=1}^{N}p(X_{i}|\pi _{i}).}
1955:
649:
3680:
611:
573:
250:
115: := 0 initialize model M(0) to represent uniform distribution over admissible solutions
6686:
6644:
6602:
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6507:
6476:
6431:
6390:
6359:
6314:
6282:
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5799:
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5593:
Towards a new evolutionary computation advances in the estimation of distribution algorithms
5539:
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270:
95:
5131:
5095:
4907:
4844:
4683:
3975:
3759:
3710:
2724:
1982:
1937:{\displaystyle D(t+1)=\alpha _{\text{CGA}}\circ S_{{\text{Sort}}(f)}\circ \beta _{2}(D(t))}
6136:
6094:
5872:
5718:
5287:
4873:
4777:
2183:
1487:
956:
181:
5930:
Larrañaga, Pedro; Karshenas, Hossein; Bielza, Concha; Santana, Roberto (21 August 2012).
2691:
variable depends on any variable in the graph (verified according to a threshold value).
1401:
3648:{\displaystyle MC=\log _{2}(\lambda +1)\sum _{\tau \in T_{\text{eCGA}}}(2^{|\tau |-1}),}
1284:
should be only slightly modified by the new solutions sampled. PBIL can be described as
4723:
4703:
4529:
4509:
4315:
4297:
4277:
4002:
3280:
1952:
variables could be modeled. A bivariate factorization can be defined as follows, where
1592:
1572:
1381:
391:
230:
210:
517:{\displaystyle D_{\text{Univariate}}:=p(X_{1},\dots ,X_{N})=\prod _{i=1}^{N}p(X_{i}).}
6722:
1802:
1238:
6708:
6624:
6583:
6543:
6336:
6220:
5912:
526:
Such factorizations are used in many different EDAs, next we describe some of them.
6453:
6412:
6263:
6177:
5955:
5682:
5472:
Stochastic hill climbing with learning by vectors of normal distributions (SHCLVND)
3440:
variables. The factorized joint probability distribution is represented as follows
2182:
Bivariate and multivariate distributions are usually represented as probabilistic
6511:
6286:
5978:
5803:
5732:
Harik, G.R.; Lobo, F.G.; Goldberg, D.E. (1999). "The compact genetic algorithm".
5534:
Pelikan, Martin (2005-02-21), "Probabilistic Model-Building Genetic Algorithms",
936:{\displaystyle D(t+1)=\alpha _{\text{UMDA}}\circ S\circ \beta _{\lambda }(D(t)).}
5543:
3531:{\displaystyle p(X_{1},\dots ,X_{N})=\prod _{\tau \in T_{\text{eCGA}}}p(\tau ).}
1609:
being the worst solution. The CGA estimates univariate probabilities as follows
6559:
6480:
6363:
6247:
6204:
6052:
6025:
5947:
5658:
4170:{\displaystyle p(X_{1},X_{2},\dots ,X_{N})=\prod _{i=1}^{N}p(X_{i}|\pi _{i}).}
2676:{\displaystyle P(t+1)=\beta _{\mu }\circ \alpha _{\text{MIMIC}}\circ S(P(t)).}
6648:
6606:
6567:
6488:
6394:
6371:
6318:
6255:
6212:
5666:
5568:
Estimation of Distribution Algorithms a New Tool for Evolutionary Computation
5445:{\displaystyle P(t+1)=R_{\text{LTGA}}(P(t))\circ \alpha _{\text{LTGA}}(P(t))}
3850:
repeated until no CCC improvements are possible and produces a linkage model
3066:{\displaystyle P(t+1)=\beta _{\mu }\circ \alpha _{\text{BMDA}}\circ S(P(t)).}
1364:{\displaystyle D(t+1)=\alpha _{\text{PIBIL}}\circ S\circ \beta _{\mu }(D(t))}
90:
distribution defined by one or more variation operators, whereas EDAs use an
6690:
6435:
5642:
3957:{\displaystyle P(t+1)=\beta _{\mu }\circ \alpha _{\text{eCGA}}\circ S(P(t))}
999:
are selected. Such individuals are then used to update the model as follows
163:
6575:
5828:
5697:
5674:
4260:{\displaystyle P(t+1)=\beta _{\mu }\circ \alpha _{\text{BOA}}\circ S(P(t))}
6159:
5969:
Thierens, Dirk (11 September 2010). "The Linkage Tree Genetic Algorithm".
295:
The most simple EDAs assume that decision variables are independent, i.e.
6519:
5932:"A review on probabilistic graphical models in evolutionary computation"
5745:
55:) concentrates around the optimum as one goes along unwinding algorithm.
6150:
Corno, Fulvio; Reorda, Matteo Sonza; Squillero, Giovanni (1998-02-27).
3267:{\displaystyle p(X_{1},\dots ,X_{N})=\prod _{i=1}^{N}p(X_{i}|\pi _{i})}
5509:
3840:{\displaystyle CPC=\lambda \sum _{\tau \in T_{\text{eCGA}}}H(\tau ).}
170:
it is possible to create an arbitrary number of candidate solutions.
6152:
The selfish gene algorithm: a new evolutionary optimization strategy
2694:
The resulting model is a forest with multiple trees rooted at nodes
43:. The illustrated example optimizes a continuous objective function
6681:
5643:"The Equation for Response to Selection and Its Use for Prediction"
5496:
Estimation multivariate normal algorithm with thresheld convergence
3364:{\displaystyle T_{\text{eCGA}}=\{\tau _{1},\dots ,\tau _{\Psi }\}}
79:
and ending with the model that generates only the global optima.
5591:
Jose A. Lozano; Larrañaga, P.; Inza, I.; Bengoetxea, E. (2006).
3541:
number of bits required to store all the marginal probabilities
103:
570:
to estimate marginal probabilities from a selected population
109:
The general procedure of an EDA is outlined in the following:
5913:"Using Machine Learning to Improve Stochastic Optimization"
5346:" terminates the algorithm and outputs the following value.
4314:. The linkage model is a linkage-tree produced stored as a
3152:{\displaystyle |\pi _{i}|\leq K,~\forall i\in 1,2,\dots ,N}
126: := generate N>0 candidate solutions by sampling M(
5931:
5616:
Pelikan, Martin; Sastry, Kumara; Cantú-Paz, Erick (2006).
4680:
indicates the transfer of the genetic material indexed by
23:
Estimation of distribution algorithm. For each iteration
4747:
Gene-pool optimal mixing Input: A family of subsets
4502:
algorithm, which work as follows. At each step the two
3707:
is the number of decision variables in the linkage set
3297:; however, it also limits the generality of such EDAs.
2317:
in relation to the true probability distribution, i.e.
2190:
Mutual information maximizing input clustering (MIMIC)
5911:
Wolpert, David H.; Rajnarayan, Dev (1 January 2013).
5499:
Dependency Structure Matrix Genetic Algorithm (DSMGA)
5481:
Compact Differential Evolution (cDE) and its variants
5361:
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5172:
5134:
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2012:
1985:
1958:
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1205:
1007:
979:
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747:
710:
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652:
614:
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540:
416:
394:
381:{\displaystyle p(X_{1},X_{2})=p(X_{1})\cdot p(X_{2})}
301:
273:
253:
233:
213:
184:
6599:
2016 IEEE Congress on Evolutionary Computation (CEC)
6311:
2011 IEEE Congress of Evolutionary Computation (CEC)
6387:
2011 IEEE Symposium on Differential Evolution (SDE)
1241:, a small value determines that the previous model
6643:, Springer Berlin Heidelberg, pp. 1620–1621,
5490:Probabilistic incremental program evolution (PIPE)
5466:Estimation of multivariate normal algorithm (EMNA)
5444:
5305:
5274:
5214:
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5120:
5085:
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6641:Genetic and Evolutionary Computation — GECCO 2003
5761:Advances in Neural Information Processing Systems
530:Univariate marginal distribution algorithm (UMDA)
6281:, Springer Berlin Heidelberg, pp. 264–273,
5493:Estimation of Gaussian networks algorithm (EGNA)
5469:Estimation of Bayesian networks algorithm (EBNA)
2686:Bivariate marginal distribution algorithm (BMDA)
2306:{\displaystyle x_{r(1)}x_{r(2)},\dots ,x_{r(N)}}
51:. The sampling (following a normal distribution
1513:is then sorted in decreasing order of fitness,
534:The UMDA is a simple EDA that uses an operator
71:probabilistic model-building genetic algorithms
6506:, Springer Berlin Heidelberg, pp. 84–92,
5487:Compact Bacterial Foraging Optimization (cBFO)
27:, a random draw is performed for a population
6544:"Probabilistic incremental program evolution"
6542:Salustowicz, null; Schmidhuber, null (1997).
6193:IEEE Transactions on Evolutionary Computation
5971:Parallel Problem Solving from Nature, PPSN XI
5734:IEEE Transactions on Evolutionary Computation
5017:{\displaystyle x_{j}\in P(t):x_{i}\neq x_{j}}
134: := evaluate all candidate solutions in
39:are then estimated using the selected points
8:
6428:2011 IEEE Workshop on Memetic Computing (MC)
5536:Hierarchical Bayesian Optimization Algorithm
4480:
4477:
4451:
4445:
4419:
4413:
4400:
4394:
4381:
4375:
4362:
4356:
4343:
4340:
3358:
3326:
2374:
2352:
853:Every UMDA step can be described as follows
174:Estimation of distribution algorithms (EDAs)
6041:Computational Optimization and Applications
6671:Hsu, Shih-Huan; Yu, Tian-Li (2015-07-11).
6236:Journal of Computer Science and Technology
5484:Compact Particle Swarm Optimization (cPSO)
2380:{\displaystyle \pi _{r(i+1)}=\{X_{r(i)}\}}
1979:contains a possible variable dependent to
1562:{\displaystyle S_{{\text{Sort}}(f)}(P(t))}
6680:
6113:
6071:
6015:
5849:
5793:
5768:
5418:
5387:
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5289:
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4779:
4758:
4752:
4725:
4705:
4685:
4641:
4620:
4614:
4593:
4587:
4563:
4551:
4531:
4511:
4498:The linkage-tree learning procedure is a
4471:
4458:
4439:
4426:
4407:
4388:
4369:
4350:
4331:
4325:
4299:
4279:
4227:
4214:
4187:
4155:
4146:
4140:
4124:
4113:
4097:
4078:
4065:
4053:
4030:
4024:
4004:
3983:
3977:
3924:
3911:
3884:
3861:
3855:
3814:
3803:
3782:
3761:
3756:is the joint entropy of the variables in
3732:
3712:
3692:
3684:
3682:
3662:
3626:
3618:
3617:
3602:
3591:
3563:
3548:
3505:
3494:
3478:
3459:
3447:
3419:
3411:
3409:
3388:
3376:
3352:
3333:
3317:
3311:
3301:Extended compact genetic algorithm (eCGA)
3282:
3255:
3246:
3240:
3224:
3213:
3197:
3178:
3166:
3102:
3096:
3087:
3085:
3030:
3017:
2990:
2959:
2950:
2944:
2931:
2919:
2906:
2901:
2885:
2872:
2860:
2847:
2842:
2826:
2807:
2788:
2782:
2759:
2753:
2732:
2726:
2705:
2699:
2640:
2627:
2600:
2553:
2544:
2529:
2516:
2500:
2489:
2467:
2454:
2438:
2419:
2400:
2394:
2359:
2328:
2322:
2288:
2260:
2241:
2235:
2203:
2158:
2149:
2143:
2127:
2116:
2100:
2081:
2062:
2056:
2028:
2022:
2013:
2011:
1990:
1984:
1963:
1957:
1910:
1887:
1886:
1873:
1846:
1821:
1810:
1768:
1705:
1692:
1670:
1657:
1641:
1622:
1616:
1594:
1574:
1525:
1524:
1518:
1489:
1450:
1429:
1403:
1383:
1337:
1318:
1291:
1265:
1252:
1246:
1204:
1142:
1108:
1093:
1075:
1062:
1031:
1012:
1006:
978:
958:
906:
887:
860:
796:
762:
746:
734:
715:
709:
677:
671:
651:
613:
575:
545:
539:
502:
486:
475:
459:
440:
421:
415:
393:
369:
347:
325:
312:
300:
272:
252:
232:
212:
183:
16:Family of stochastic optimization methods
6279:Applications of Evolutionary Computation
5566:Pedro Larrañaga; Jose A. Lozano (2002).
3397:{\displaystyle \tau \in T_{\text{eCGA}}}
2983:Each step of BMDA is defined as follows
18:
5526:
6132:
6121:
6090:
6079:
5868:
5857:
5714:
5703:
5641:Mühlenbein, Heinz (1 September 1997).
5215:{\displaystyle f(x_{i})\leq f_{x_{i}}}
4572:{\displaystyle \tau \in T_{\text{LT}}}
408:univariate probability distributions,
94:probability distribution encoded by a
4270:Linkage-tree Genetic Algorithm (LTGA)
3967:Bayesian optimization algorithm (BOA)
1477:{\displaystyle P(t)=\beta _{2}(D(t))}
947:Population-based incremental learning
61:Estimation of distribution algorithms
7:
3371:is a set of subsets, in which every
5890:(1st ed.). Berlin : Springer.
3999:can be conditioned on a maximum of
6504:Swarm and Evolutionary Computation
5696:Baluja, Shummet (1 January 1994).
5463:Hill climbing with learning (HCwL)
3353:
3119:
2857:
2702:
1718:
1154:
808:
14:
3677:is the selected population size,
992:{\displaystyle \lambda \leq \mu }
1794:{\displaystyle \gamma \in (0,1]}
1230:{\displaystyle \gamma \in (0,1]}
100:multivariate normal distribution
4629:{\displaystyle R_{\text{LTGA}}}
3870:{\displaystyle T_{\text{eCGA}}}
1717:
1374:Compact genetic algorithm (cGA)
119:(termination criteria not met)
5833:(phd). University of Michigan.
5439:
5436:
5430:
5424:
5408:
5405:
5399:
5393:
5377:
5365:
5300:
5294:
5269:
5263:
5247:
5241:
5189:
5176:
5151:
5145:
5115:
5109:
5080:
5067:
4985:
4979:
4886:
4880:
4825:
4813:
4790:
4784:
4667:
4661:
4655:
4652:
4646:
4254:
4251:
4245:
4239:
4204:
4192:
4161:
4147:
4133:
4103:
4058:
3951:
3948:
3942:
3936:
3901:
3889:
3831:
3825:
3743:
3737:
3693:
3685:
3639:
3627:
3619:
3610:
3584:
3572:
3522:
3516:
3484:
3452:
3420:
3412:
3261:
3247:
3233:
3203:
3171:
3103:
3088:
3057:
3054:
3048:
3042:
3007:
2995:
2965:
2951:
2937:
2891:
2878:
2832:
2800:
2667:
2664:
2658:
2652:
2617:
2605:
2574:
2569:
2557:
2545:
2539:
2533:
2522:
2482:
2477:
2471:
2460:
2444:
2412:
2387:. MIMIC models a distribution
2369:
2363:
2344:
2332:
2298:
2292:
2270:
2264:
2251:
2245:
2214:
2196:joint probability distribution
2164:
2150:
2136:
2106:
2074:
2029:
2014:
1931:
1928:
1922:
1916:
1898:
1892:
1863:
1851:
1788:
1776:
1711:
1685:
1676:
1663:
1647:
1634:
1556:
1553:
1547:
1541:
1536:
1530:
1500:
1494:
1471:
1468:
1462:
1456:
1440:
1434:
1358:
1355:
1349:
1343:
1308:
1296:
1271:
1258:
1224:
1212:
1133:
1130:
1124:
1118:
1101:
1087:
1081:
1068:
1055:
1043:
1037:
1024:
927:
924:
918:
912:
877:
865:
787:
784:
778:
772:
740:
727:
695:{\displaystyle \alpha _{UMDA}}
633:
630:
624:
618:
595:
592:
586:
580:
563:{\displaystyle \alpha _{UMDA}}
508:
495:
465:
433:
375:
362:
353:
340:
331:
305:
194:
188:
138:M(t + 1) := adjust_model(
35:. The distribution parameters
1:
5700:. Carnegie Mellon University.
4946:{\displaystyle T_{\text{LT}}}
4767:{\displaystyle T_{\text{LT}}}
4602:{\displaystyle T_{\text{LT}}}
3433:{\displaystyle |\tau |\leq K}
3404:is a linkage set, containing
2714:{\displaystyle \Upsilon _{t}}
6601:. IEEE. pp. 3425–3432.
6512:10.1007/978-3-642-29353-5_10
6313:. IEEE. pp. 1972–1977.
6287:10.1007/978-3-642-20525-5_27
5979:10.1007/978-3-642-15844-5_27
5827:Harik, Georges Raif (1997).
5804:10.1007/978-1-4471-0819-1_39
5460:Probability collectives (PC)
5275:{\displaystyle x_{i}:=x_{j}}
4019:other variables, defined in
2223:{\displaystyle r:i\mapsto j}
2042:{\displaystyle |\pi _{i}|=1}
1839:. The CGA can be defined as
1277:{\displaystyle p_{t}(X_{i})}
1237:is a parameter defining the
973:individuals are sampled and
247:, a model-building operator
82:EDAs belong to the class of
6004:Advances in Complex Systems
5570:. Boston, MA: Springer US.
5544:10.1007/978-3-540-32373-0_2
5478:Selfish Gene Algorithm (SG)
3076:Multivariate factorizations
2315:Kullback-Leibler divergence
1832:{\displaystyle \gamma =1/N}
1801:is a constant defining the
6750:
5786:Advances in Soft Computing
5329:" means that the value of
6675:. ACM. pp. 519–526.
6560:10.1162/evco.1997.5.2.123
6481:10.1016/j.ins.2013.03.026
6364:10.1016/j.ins.2011.02.004
6248:10.1007/s11390-012-1284-2
6205:10.1109/tevc.2010.2058120
6154:. ACM. pp. 349–355.
6026:10.1142/S0219525906000884
5948:10.1007/s10732-012-9208-4
5659:10.1162/evco.1997.5.3.303
5050:{\displaystyle f_{x_{i}}}
2194:The MIMIC factorizes the
291:Univariate factorizations
6729:Evolutionary computation
6649:10.1007/3-540-45110-2_54
6607:10.1109/cec.2016.7744223
6548:Evolutionary Computation
6395:10.1109/sde.2011.5952051
6319:10.1109/cec.2011.5949857
5919:. Aaaiws'13-17: 146–148.
5886:Pelikan, Martin (2005).
5333:changes to the value of
5086:{\displaystyle f(x_{i})}
4673:{\displaystyle x\gets y}
4039:{\displaystyle \pi _{i}}
3749:{\displaystyle H(\tau )}
3670:{\displaystyle \lambda }
2768:{\displaystyle \pi _{i}}
1972:{\displaystyle \pi _{i}}
1947:Bivariate factorizations
702:produces probabilities:
659:{\displaystyle \lambda }
267:and a sampling operator
6734:Stochastic optimization
6691:10.1145/2739480.2754737
6436:10.1109/mc.2011.5953633
6053:10.1023/A:1013500812258
4500:hierarchical clustering
3700:{\displaystyle |\tau |}
639:{\displaystyle S(P(t))}
601:{\displaystyle S(P(t))}
260:{\displaystyle \alpha }
227:, a selection operator
84:evolutionary algorithms
76:stochastic optimization
6430:. IEEE. pp. 1–8.
6389:. IEEE. pp. 1–8.
6131:Cite journal requires
6089:Cite journal requires
5867:Cite journal requires
5713:Cite journal requires
5446:
5307:
5276:
5216:
5158:
5122:
5087:
5051:
5018:
4947:
4918:
4893:
4862:
4832:
4831:{\displaystyle P(t+1)}
4797:
4768:
4734:
4714:
4694:
4674:
4630:
4603:
4573:
4540:
4520:
4490:
4308:
4288:
4261:
4171:
4129:
4040:
4013:
3993:
3958:
3871:
3841:
3770:
3750:
3721:
3701:
3671:
3649:
3532:
3434:
3398:
3365:
3291:
3268:
3229:
3153:
3067:
2975:
2769:
2742:
2715:
2677:
2584:
2511:
2381:
2307:
2224:
2174:
2132:
2043:
2000:
1973:
1938:
1833:
1795:
1755:
1603:
1583:
1563:
1507:
1478:
1418:
1392:
1365:
1278:
1231:
1191:
993:
967:
937:
845:
696:
660:
640:
602:
564:
518:
491:
402:
382:
281:
280:{\displaystyle \beta }
261:
241:
221:
201:
56:
47:with a unique optimum
6160:10.1145/330560.330838
5936:Journal of Heuristics
5447:
5308:
5277:
5217:
5159:
5157:{\displaystyle x_{j}}
5123:
5121:{\displaystyle x_{i}}
5088:
5052:
5019:
4948:
4919:
4917:{\displaystyle \tau }
4894:
4863:
4861:{\displaystyle x_{i}}
4833:
4803:Output: A population
4798:
4769:
4735:
4715:
4695:
4693:{\displaystyle \tau }
4675:
4636:, where the notation
4631:
4604:
4574:
4541:
4521:
4491:
4309:
4289:
4262:
4172:
4109:
4041:
4014:
3994:
3992:{\displaystyle x_{i}}
3959:
3872:
3842:
3771:
3769:{\displaystyle \tau }
3751:
3722:
3720:{\displaystyle \tau }
3702:
3672:
3650:
3533:
3435:
3399:
3366:
3292:
3269:
3209:
3154:
3068:
2976:
2770:
2743:
2741:{\displaystyle I_{t}}
2716:
2678:
2585:
2485:
2382:
2308:
2225:
2175:
2112:
2044:
2001:
1999:{\displaystyle X_{i}}
1974:
1939:
1834:
1796:
1756:
1604:
1584:
1564:
1508:
1479:
1419:
1393:
1366:
1279:
1232:
1192:
994:
968:
938:
846:
697:
661:
641:
603:
565:
519:
471:
403:
383:
282:
262:
242:
222:
202:
22:
6469:Information Sciences
6352:Information Sciences
5973:. pp. 264–273.
5788:. pp. 521–535.
5620:. Berlin: Springer.
5595:. Berlin: Springer.
5515:Cross-entropy method
5359:
5306:{\displaystyle P(t)}
5288:
5228:
5170:
5132:
5096:
5061:
5027:
4960:
4930:
4908:
4892:{\displaystyle P(t)}
4874:
4845:
4807:
4796:{\displaystyle P(t)}
4778:
4751:
4724:
4704:
4684:
4640:
4613:
4586:
4550:
4530:
4510:
4324:
4298:
4278:
4186:
4052:
4023:
4003:
3976:
3883:
3854:
3781:
3760:
3731:
3711:
3681:
3661:
3547:
3446:
3408:
3375:
3310:
3281:
3165:
3084:
2989:
2781:
2752:
2725:
2698:
2599:
2393:
2321:
2234:
2202:
2055:
2010:
1983:
1956:
1845:
1809:
1767:
1615:
1593:
1573:
1517:
1506:{\displaystyle P(t)}
1488:
1428:
1402:
1382:
1290:
1245:
1203:
1005:
977:
966:{\displaystyle \mu }
957:
859:
708:
670:
650:
612:
574:
538:
414:
392:
299:
271:
251:
231:
211:
200:{\displaystyle P(t)}
182:
68:), sometimes called
5746:10.1109/4235.797971
1589:being the best and
1417:{\displaystyle x,y}
5442:
5321:. For instance, "
5317:"←" denotes
5303:
5272:
5212:
5154:
5118:
5083:
5047:
5014:
4943:
4914:
4889:
4858:
4828:
4793:
4764:
4730:
4710:
4690:
4670:
4626:
4599:
4569:
4536:
4516:
4486:
4304:
4284:
4257:
4167:
4036:
4009:
3989:
3954:
3867:
3837:
3821:
3766:
3746:
3717:
3697:
3667:
3645:
3609:
3528:
3512:
3430:
3394:
3361:
3287:
3264:
3149:
3063:
2971:
2926:
2867:
2765:
2738:
2711:
2673:
2580:
2377:
2303:
2220:
2170:
2039:
1996:
1969:
1934:
1829:
1791:
1751:
1599:
1579:
1559:
1503:
1474:
1414:
1398:, two individuals
1388:
1361:
1274:
1227:
1187:
1137:
989:
963:
933:
841:
791:
756:
692:
656:
636:
598:
560:
514:
398:
378:
277:
257:
237:
217:
207:at the generation
197:
57:
31:in a distribution
6358:(12): 2469–2487.
5988:978-3-642-15843-8
5897:978-3-540-23774-7
5813:978-1-85233-062-0
5647:Evol. Computation
5602:978-3-540-32494-2
5577:978-1-4615-1539-5
5421:
5390:
5347:
5338:
4940:
4774:and a population
4761:
4733:{\displaystyle x}
4713:{\displaystyle y}
4623:
4596:
4566:
4539:{\displaystyle j}
4519:{\displaystyle i}
4334:
4307:{\displaystyle T}
4287:{\displaystyle T}
4230:
4012:{\displaystyle K}
3927:
3864:
3817:
3799:
3605:
3587:
3508:
3490:
3391:
3320:
3290:{\displaystyle N}
3118:
3033:
2897:
2838:
2643:
2065:
1890:
1876:
1805:, usually set to
1602:{\displaystyle v}
1582:{\displaystyle u}
1528:
1484:. The population
1391:{\displaystyle t}
1321:
1153:
1104:
890:
807:
758:
755:
424:
401:{\displaystyle N}
240:{\displaystyle S}
220:{\displaystyle t}
6741:
6713:
6712:
6684:
6668:
6662:
6661:
6635:
6629:
6628:
6594:
6588:
6587:
6539:
6533:
6532:
6499:
6493:
6492:
6464:
6458:
6457:
6423:
6417:
6416:
6382:
6376:
6375:
6347:
6341:
6340:
6306:
6300:
6299:
6274:
6268:
6267:
6242:(5): 1056–1076.
6231:
6225:
6224:
6188:
6182:
6181:
6147:
6141:
6140:
6134:
6129:
6127:
6119:
6117:
6105:
6099:
6098:
6092:
6087:
6085:
6077:
6075:
6063:
6057:
6056:
6036:
6030:
6029:
6019:
5999:
5993:
5992:
5966:
5960:
5959:
5927:
5921:
5920:
5908:
5902:
5901:
5883:
5877:
5876:
5870:
5865:
5863:
5855:
5853:
5841:
5835:
5834:
5824:
5818:
5817:
5797:
5781:
5775:
5774:
5772:
5756:
5750:
5749:
5729:
5723:
5722:
5716:
5711:
5709:
5701:
5693:
5687:
5686:
5638:
5632:
5631:
5613:
5607:
5606:
5588:
5582:
5581:
5563:
5557:
5556:
5531:
5451:
5449:
5448:
5443:
5423:
5422:
5419:
5392:
5391:
5388:
5341:
5316:
5312:
5310:
5309:
5304:
5281:
5279:
5278:
5273:
5262:
5261:
5240:
5239:
5221:
5219:
5218:
5213:
5211:
5210:
5209:
5208:
5188:
5187:
5163:
5161:
5160:
5155:
5144:
5143:
5127:
5125:
5124:
5119:
5108:
5107:
5092:
5090:
5089:
5084:
5079:
5078:
5056:
5054:
5053:
5048:
5046:
5045:
5044:
5043:
5023:
5021:
5020:
5015:
5013:
5012:
5000:
4999:
4972:
4971:
4956:choose a random
4952:
4950:
4949:
4944:
4942:
4941:
4938:
4923:
4921:
4920:
4915:
4898:
4896:
4895:
4890:
4867:
4865:
4864:
4859:
4857:
4856:
4837:
4835:
4834:
4829:
4802:
4800:
4799:
4794:
4773:
4771:
4770:
4765:
4763:
4762:
4759:
4739:
4737:
4736:
4731:
4719:
4717:
4716:
4711:
4699:
4697:
4696:
4691:
4679:
4677:
4676:
4671:
4635:
4633:
4632:
4627:
4625:
4624:
4621:
4608:
4606:
4605:
4600:
4598:
4597:
4594:
4578:
4576:
4575:
4570:
4568:
4567:
4564:
4545:
4543:
4542:
4537:
4525:
4523:
4522:
4517:
4495:
4493:
4492:
4487:
4476:
4475:
4463:
4462:
4444:
4443:
4431:
4430:
4412:
4411:
4393:
4392:
4374:
4373:
4355:
4354:
4336:
4335:
4332:
4313:
4311:
4310:
4305:
4293:
4291:
4290:
4285:
4266:
4264:
4263:
4258:
4232:
4231:
4228:
4219:
4218:
4176:
4174:
4173:
4168:
4160:
4159:
4150:
4145:
4144:
4128:
4123:
4102:
4101:
4083:
4082:
4070:
4069:
4045:
4043:
4042:
4037:
4035:
4034:
4018:
4016:
4015:
4010:
3998:
3996:
3995:
3990:
3988:
3987:
3963:
3961:
3960:
3955:
3929:
3928:
3925:
3916:
3915:
3876:
3874:
3873:
3868:
3866:
3865:
3862:
3846:
3844:
3843:
3838:
3820:
3819:
3818:
3815:
3775:
3773:
3772:
3767:
3755:
3753:
3752:
3747:
3726:
3724:
3723:
3718:
3706:
3704:
3703:
3698:
3696:
3688:
3676:
3674:
3673:
3668:
3654:
3652:
3651:
3646:
3638:
3637:
3630:
3622:
3608:
3607:
3606:
3603:
3568:
3567:
3537:
3535:
3534:
3529:
3511:
3510:
3509:
3506:
3483:
3482:
3464:
3463:
3439:
3437:
3436:
3431:
3423:
3415:
3403:
3401:
3400:
3395:
3393:
3392:
3389:
3370:
3368:
3367:
3362:
3357:
3356:
3338:
3337:
3322:
3321:
3318:
3296:
3294:
3293:
3288:
3273:
3271:
3270:
3265:
3260:
3259:
3250:
3245:
3244:
3228:
3223:
3202:
3201:
3183:
3182:
3158:
3156:
3155:
3150:
3116:
3106:
3101:
3100:
3091:
3072:
3070:
3069:
3064:
3035:
3034:
3031:
3022:
3021:
2980:
2978:
2977:
2972:
2964:
2963:
2954:
2949:
2948:
2936:
2935:
2925:
2924:
2923:
2911:
2910:
2890:
2889:
2877:
2876:
2866:
2865:
2864:
2852:
2851:
2831:
2830:
2812:
2811:
2799:
2798:
2774:
2772:
2771:
2766:
2764:
2763:
2747:
2745:
2744:
2739:
2737:
2736:
2720:
2718:
2717:
2712:
2710:
2709:
2682:
2680:
2679:
2674:
2645:
2644:
2641:
2632:
2631:
2589:
2587:
2586:
2581:
2573:
2572:
2548:
2543:
2542:
2521:
2520:
2510:
2499:
2481:
2480:
2459:
2458:
2443:
2442:
2424:
2423:
2411:
2410:
2386:
2384:
2383:
2378:
2373:
2372:
2348:
2347:
2312:
2310:
2309:
2304:
2302:
2301:
2274:
2273:
2255:
2254:
2229:
2227:
2226:
2221:
2184:graphical models
2179:
2177:
2176:
2171:
2163:
2162:
2153:
2148:
2147:
2131:
2126:
2105:
2104:
2086:
2085:
2067:
2066:
2063:
2048:
2046:
2045:
2040:
2032:
2027:
2026:
2017:
2005:
2003:
2002:
1997:
1995:
1994:
1978:
1976:
1975:
1970:
1968:
1967:
1943:
1941:
1940:
1935:
1915:
1914:
1902:
1901:
1891:
1888:
1878:
1877:
1874:
1838:
1836:
1835:
1830:
1825:
1800:
1798:
1797:
1792:
1760:
1758:
1757:
1752:
1710:
1709:
1697:
1696:
1675:
1674:
1662:
1661:
1646:
1645:
1633:
1632:
1608:
1606:
1605:
1600:
1588:
1586:
1585:
1580:
1568:
1566:
1565:
1560:
1540:
1539:
1529:
1526:
1512:
1510:
1509:
1504:
1483:
1481:
1480:
1475:
1455:
1454:
1423:
1421:
1420:
1415:
1397:
1395:
1394:
1389:
1370:
1368:
1367:
1362:
1342:
1341:
1323:
1322:
1319:
1283:
1281:
1280:
1275:
1270:
1269:
1257:
1256:
1236:
1234:
1233:
1228:
1196:
1194:
1193:
1188:
1151:
1147:
1146:
1136:
1097:
1080:
1079:
1067:
1066:
1036:
1035:
1023:
1022:
998:
996:
995:
990:
972:
970:
969:
964:
942:
940:
939:
934:
911:
910:
892:
891:
888:
850:
848:
847:
842:
805:
801:
800:
790:
757:
748:
739:
738:
726:
725:
701:
699:
698:
693:
691:
690:
665:
663:
662:
657:
645:
643:
642:
637:
607:
605:
604:
599:
569:
567:
566:
561:
559:
558:
523:
521:
520:
515:
507:
506:
490:
485:
464:
463:
445:
444:
426:
425:
422:
407:
405:
404:
399:
387:
385:
384:
379:
374:
373:
352:
351:
330:
329:
317:
316:
286:
284:
283:
278:
266:
264:
263:
258:
246:
244:
243:
238:
226:
224:
223:
218:
206:
204:
203:
198:
96:Bayesian network
6749:
6748:
6744:
6743:
6742:
6740:
6739:
6738:
6719:
6718:
6717:
6716:
6701:
6670:
6669:
6665:
6659:
6637:
6636:
6632:
6617:
6596:
6595:
6591:
6541:
6540:
6536:
6530:
6501:
6500:
6496:
6466:
6465:
6461:
6446:
6425:
6424:
6420:
6405:
6384:
6383:
6379:
6349:
6348:
6344:
6329:
6308:
6307:
6303:
6297:
6276:
6275:
6271:
6233:
6232:
6228:
6190:
6189:
6185:
6170:
6149:
6148:
6144:
6130:
6120:
6107:
6106:
6102:
6088:
6078:
6065:
6064:
6060:
6038:
6037:
6033:
6017:10.1.1.154.6395
6001:
6000:
5996:
5989:
5968:
5967:
5963:
5929:
5928:
5924:
5910:
5909:
5905:
5898:
5885:
5884:
5880:
5866:
5856:
5843:
5842:
5838:
5826:
5825:
5821:
5814:
5783:
5782:
5778:
5758:
5757:
5753:
5731:
5730:
5726:
5712:
5702:
5695:
5694:
5690:
5640:
5639:
5635:
5628:
5615:
5614:
5610:
5603:
5590:
5589:
5585:
5578:
5565:
5564:
5560:
5554:
5533:
5532:
5528:
5523:
5506:
5475:Real-coded PBIL
5457:
5414:
5383:
5357:
5356:
5350:
5313:
5286:
5285:
5253:
5231:
5226:
5225:
5200:
5195:
5179:
5168:
5167:
5135:
5130:
5129:
5099:
5094:
5093:
5070:
5059:
5058:
5035:
5030:
5025:
5024:
5004:
4991:
4963:
4958:
4957:
4933:
4928:
4927:
4906:
4905:
4872:
4871:
4848:
4843:
4842:
4805:
4804:
4776:
4775:
4754:
4749:
4748:
4722:
4721:
4702:
4701:
4682:
4681:
4638:
4637:
4616:
4611:
4610:
4589:
4584:
4583:
4559:
4548:
4547:
4528:
4527:
4508:
4507:
4467:
4454:
4435:
4422:
4403:
4384:
4365:
4346:
4327:
4322:
4321:
4296:
4295:
4276:
4275:
4272:
4223:
4210:
4184:
4183:
4151:
4136:
4093:
4074:
4061:
4050:
4049:
4026:
4021:
4020:
4001:
4000:
3979:
3974:
3973:
3969:
3920:
3907:
3881:
3880:
3857:
3852:
3851:
3810:
3779:
3778:
3758:
3757:
3729:
3728:
3709:
3708:
3679:
3678:
3659:
3658:
3613:
3598:
3559:
3545:
3544:
3501:
3474:
3455:
3444:
3443:
3406:
3405:
3384:
3373:
3372:
3348:
3329:
3313:
3308:
3307:
3303:
3279:
3278:
3251:
3236:
3193:
3174:
3163:
3162:
3092:
3082:
3081:
3078:
3026:
3013:
2987:
2986:
2955:
2940:
2927:
2915:
2902:
2881:
2868:
2856:
2843:
2822:
2803:
2784:
2779:
2778:
2755:
2750:
2749:
2728:
2723:
2722:
2701:
2696:
2695:
2688:
2636:
2623:
2597:
2596:
2549:
2525:
2512:
2463:
2450:
2434:
2415:
2396:
2391:
2390:
2355:
2324:
2319:
2318:
2284:
2256:
2237:
2232:
2231:
2200:
2199:
2192:
2154:
2139:
2096:
2077:
2058:
2053:
2052:
2018:
2008:
2007:
1986:
1981:
1980:
1959:
1954:
1953:
1949:
1906:
1882:
1869:
1843:
1842:
1807:
1806:
1765:
1764:
1701:
1688:
1666:
1653:
1637:
1618:
1613:
1612:
1591:
1590:
1571:
1570:
1520:
1515:
1514:
1486:
1485:
1446:
1426:
1425:
1400:
1399:
1380:
1379:
1376:
1333:
1314:
1288:
1287:
1261:
1248:
1243:
1242:
1201:
1200:
1138:
1071:
1058:
1027:
1008:
1003:
1002:
975:
974:
955:
954:
951:
902:
883:
857:
856:
792:
730:
711:
706:
705:
673:
668:
667:
648:
647:
610:
609:
572:
571:
541:
536:
535:
532:
498:
455:
436:
417:
412:
411:
390:
389:
365:
343:
321:
308:
297:
296:
293:
269:
268:
249:
248:
229:
228:
209:
208:
180:
179:
176:
159:
17:
12:
11:
5:
6747:
6745:
6737:
6736:
6731:
6721:
6720:
6715:
6714:
6699:
6663:
6657:
6630:
6615:
6589:
6554:(2): 123–141.
6534:
6528:
6494:
6459:
6444:
6418:
6403:
6377:
6342:
6327:
6301:
6295:
6269:
6226:
6183:
6169:978-0897919692
6168:
6142:
6133:|journal=
6115:10.1.1.19.3536
6100:
6091:|journal=
6073:10.1.1.19.3536
6058:
6031:
6010:(4): 383–436.
5994:
5987:
5961:
5942:(5): 795–819.
5922:
5903:
5896:
5878:
5869:|journal=
5851:10.1.1.46.8131
5836:
5819:
5812:
5795:10.1.1.55.1151
5776:
5770:10.1.1.47.6497
5751:
5740:(4): 287–297.
5724:
5715:|journal=
5688:
5653:(3): 303–346.
5633:
5627:978-3540349532
5626:
5608:
5601:
5583:
5576:
5558:
5552:
5525:
5524:
5522:
5519:
5518:
5517:
5512:
5505:
5502:
5501:
5500:
5497:
5494:
5491:
5488:
5485:
5482:
5479:
5476:
5473:
5470:
5467:
5464:
5461:
5456:
5453:
5441:
5438:
5435:
5432:
5429:
5426:
5417:
5413:
5410:
5407:
5404:
5401:
5398:
5395:
5386:
5382:
5379:
5376:
5373:
5370:
5367:
5364:
5349:
5348:
5339:
5302:
5299:
5296:
5293:
5271:
5268:
5265:
5260:
5256:
5252:
5249:
5246:
5243:
5238:
5234:
5207:
5203:
5198:
5194:
5191:
5186:
5182:
5178:
5175:
5153:
5150:
5147:
5142:
5138:
5117:
5114:
5111:
5106:
5102:
5082:
5077:
5073:
5069:
5066:
5042:
5038:
5033:
5011:
5007:
5003:
4998:
4994:
4990:
4987:
4984:
4981:
4978:
4975:
4970:
4966:
4936:
4913:
4888:
4885:
4882:
4879:
4855:
4851:
4827:
4824:
4821:
4818:
4815:
4812:
4792:
4789:
4786:
4783:
4757:
4743:
4742:
4729:
4709:
4689:
4669:
4666:
4663:
4660:
4657:
4654:
4651:
4648:
4645:
4619:
4592:
4582:The LTGA uses
4562:
4558:
4555:
4535:
4515:
4485:
4482:
4479:
4474:
4470:
4466:
4461:
4457:
4453:
4450:
4447:
4442:
4438:
4434:
4429:
4425:
4421:
4418:
4415:
4410:
4406:
4402:
4399:
4396:
4391:
4387:
4383:
4380:
4377:
4372:
4368:
4364:
4361:
4358:
4353:
4349:
4345:
4342:
4339:
4330:
4316:Family of sets
4303:
4283:
4271:
4268:
4256:
4253:
4250:
4247:
4244:
4241:
4238:
4235:
4226:
4222:
4217:
4213:
4209:
4206:
4203:
4200:
4197:
4194:
4191:
4166:
4163:
4158:
4154:
4149:
4143:
4139:
4135:
4132:
4127:
4122:
4119:
4116:
4112:
4108:
4105:
4100:
4096:
4092:
4089:
4086:
4081:
4077:
4073:
4068:
4064:
4060:
4057:
4033:
4029:
4008:
3986:
3982:
3968:
3965:
3953:
3950:
3947:
3944:
3941:
3938:
3935:
3932:
3923:
3919:
3914:
3910:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3860:
3836:
3833:
3830:
3827:
3824:
3813:
3809:
3806:
3802:
3798:
3795:
3792:
3789:
3786:
3765:
3745:
3742:
3739:
3736:
3716:
3695:
3691:
3687:
3666:
3644:
3641:
3636:
3633:
3629:
3625:
3621:
3616:
3612:
3601:
3597:
3594:
3590:
3586:
3583:
3580:
3577:
3574:
3571:
3566:
3562:
3558:
3555:
3552:
3527:
3524:
3521:
3518:
3515:
3504:
3500:
3497:
3493:
3489:
3486:
3481:
3477:
3473:
3470:
3467:
3462:
3458:
3454:
3451:
3429:
3426:
3422:
3418:
3414:
3387:
3383:
3380:
3360:
3355:
3351:
3347:
3344:
3341:
3336:
3332:
3328:
3325:
3316:
3302:
3299:
3286:
3263:
3258:
3254:
3249:
3243:
3239:
3235:
3232:
3227:
3222:
3219:
3216:
3212:
3208:
3205:
3200:
3196:
3192:
3189:
3186:
3181:
3177:
3173:
3170:
3148:
3145:
3142:
3139:
3136:
3133:
3130:
3127:
3124:
3121:
3115:
3112:
3109:
3105:
3099:
3095:
3090:
3077:
3074:
3062:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3029:
3025:
3020:
3016:
3012:
3009:
3006:
3003:
3000:
2997:
2994:
2970:
2967:
2962:
2958:
2953:
2947:
2943:
2939:
2934:
2930:
2922:
2918:
2914:
2909:
2905:
2900:
2896:
2893:
2888:
2884:
2880:
2875:
2871:
2863:
2859:
2855:
2850:
2846:
2841:
2837:
2834:
2829:
2825:
2821:
2818:
2815:
2810:
2806:
2802:
2797:
2794:
2791:
2787:
2762:
2758:
2735:
2731:
2721:. Considering
2708:
2704:
2687:
2684:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2648:
2639:
2635:
2630:
2626:
2622:
2619:
2616:
2613:
2610:
2607:
2604:
2579:
2576:
2571:
2568:
2565:
2562:
2559:
2556:
2552:
2547:
2541:
2538:
2535:
2532:
2528:
2524:
2519:
2515:
2509:
2506:
2503:
2498:
2495:
2492:
2488:
2484:
2479:
2476:
2473:
2470:
2466:
2462:
2457:
2453:
2449:
2446:
2441:
2437:
2433:
2430:
2427:
2422:
2418:
2414:
2409:
2406:
2403:
2399:
2376:
2371:
2368:
2365:
2362:
2358:
2354:
2351:
2346:
2343:
2340:
2337:
2334:
2331:
2327:
2313:minimizes the
2300:
2297:
2294:
2291:
2287:
2283:
2280:
2277:
2272:
2269:
2266:
2263:
2259:
2253:
2250:
2247:
2244:
2240:
2219:
2216:
2213:
2210:
2207:
2191:
2188:
2169:
2166:
2161:
2157:
2152:
2146:
2142:
2138:
2135:
2130:
2125:
2122:
2119:
2115:
2111:
2108:
2103:
2099:
2095:
2092:
2089:
2084:
2080:
2076:
2073:
2070:
2061:
2038:
2035:
2031:
2025:
2021:
2016:
1993:
1989:
1966:
1962:
1948:
1945:
1933:
1930:
1927:
1924:
1921:
1918:
1913:
1909:
1905:
1900:
1897:
1894:
1885:
1881:
1872:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1828:
1824:
1820:
1817:
1814:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1716:
1713:
1708:
1704:
1700:
1695:
1691:
1687:
1684:
1681:
1678:
1673:
1669:
1665:
1660:
1656:
1652:
1649:
1644:
1640:
1636:
1631:
1628:
1625:
1621:
1598:
1578:
1558:
1555:
1552:
1549:
1546:
1543:
1538:
1535:
1532:
1523:
1502:
1499:
1496:
1493:
1473:
1470:
1467:
1464:
1461:
1458:
1453:
1449:
1445:
1442:
1439:
1436:
1433:
1413:
1410:
1407:
1387:
1375:
1372:
1360:
1357:
1354:
1351:
1348:
1345:
1340:
1336:
1332:
1329:
1326:
1317:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1273:
1268:
1264:
1260:
1255:
1251:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1150:
1145:
1141:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1107:
1103:
1100:
1096:
1092:
1089:
1086:
1083:
1078:
1074:
1070:
1065:
1061:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1034:
1030:
1026:
1021:
1018:
1015:
1011:
988:
985:
982:
962:
950:
944:
932:
929:
926:
923:
920:
917:
914:
909:
905:
901:
898:
895:
886:
882:
879:
876:
873:
870:
867:
864:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
810:
804:
799:
795:
789:
786:
783:
780:
777:
774:
771:
768:
765:
761:
754:
751:
745:
742:
737:
733:
729:
724:
721:
718:
714:
689:
686:
683:
680:
676:
655:
635:
632:
629:
626:
623:
620:
617:
608:. By assuming
597:
594:
591:
588:
585:
582:
579:
557:
554:
551:
548:
544:
531:
528:
513:
510:
505:
501:
497:
494:
489:
484:
481:
478:
474:
470:
467:
462:
458:
454:
451:
448:
443:
439:
435:
432:
429:
420:
397:
377:
372:
368:
364:
361:
358:
355:
350:
346:
342:
339:
336:
333:
328:
324:
320:
315:
311:
307:
304:
292:
289:
276:
256:
236:
216:
196:
193:
190:
187:
175:
172:
111:
74:(PMBGAs), are
15:
13:
10:
9:
6:
4:
3:
2:
6746:
6735:
6732:
6730:
6727:
6726:
6724:
6710:
6706:
6702:
6700:9781450334723
6696:
6692:
6688:
6683:
6678:
6674:
6667:
6664:
6660:
6658:9783540406037
6654:
6650:
6646:
6642:
6634:
6631:
6626:
6622:
6618:
6616:9781509006236
6612:
6608:
6604:
6600:
6593:
6590:
6585:
6581:
6577:
6573:
6569:
6565:
6561:
6557:
6553:
6549:
6545:
6538:
6535:
6531:
6529:9783642293528
6525:
6521:
6517:
6513:
6509:
6505:
6498:
6495:
6490:
6486:
6482:
6478:
6474:
6470:
6463:
6460:
6455:
6451:
6447:
6445:9781612840659
6441:
6437:
6433:
6429:
6422:
6419:
6414:
6410:
6406:
6404:9781612840710
6400:
6396:
6392:
6388:
6381:
6378:
6373:
6369:
6365:
6361:
6357:
6353:
6346:
6343:
6338:
6334:
6330:
6328:9781424478347
6324:
6320:
6316:
6312:
6305:
6302:
6298:
6296:9783642205248
6292:
6288:
6284:
6280:
6273:
6270:
6265:
6261:
6257:
6253:
6249:
6245:
6241:
6237:
6230:
6227:
6222:
6218:
6214:
6210:
6206:
6202:
6198:
6194:
6187:
6184:
6179:
6175:
6171:
6165:
6161:
6157:
6153:
6146:
6143:
6138:
6125:
6116:
6111:
6104:
6101:
6096:
6083:
6074:
6069:
6062:
6059:
6054:
6050:
6046:
6042:
6035:
6032:
6027:
6023:
6018:
6013:
6009:
6005:
5998:
5995:
5990:
5984:
5980:
5976:
5972:
5965:
5962:
5957:
5953:
5949:
5945:
5941:
5937:
5933:
5926:
5923:
5918:
5914:
5907:
5904:
5899:
5893:
5889:
5882:
5879:
5874:
5861:
5852:
5847:
5840:
5837:
5832:
5831:
5823:
5820:
5815:
5809:
5805:
5801:
5796:
5791:
5787:
5780:
5777:
5771:
5766:
5762:
5755:
5752:
5747:
5743:
5739:
5735:
5728:
5725:
5720:
5707:
5699:
5692:
5689:
5684:
5680:
5676:
5672:
5668:
5664:
5660:
5656:
5652:
5648:
5644:
5637:
5634:
5629:
5623:
5619:
5612:
5609:
5604:
5598:
5594:
5587:
5584:
5579:
5573:
5569:
5562:
5559:
5555:
5553:9783540237747
5549:
5545:
5541:
5537:
5530:
5527:
5520:
5516:
5513:
5511:
5508:
5507:
5503:
5498:
5495:
5492:
5489:
5486:
5483:
5480:
5477:
5474:
5471:
5468:
5465:
5462:
5459:
5458:
5454:
5452:
5433:
5427:
5415:
5411:
5402:
5396:
5384:
5380:
5374:
5371:
5368:
5362:
5354:
5345:
5340:
5336:
5332:
5328:
5324:
5320:
5315:
5314:
5297:
5291:
5284:
5266:
5258:
5254:
5250:
5244:
5236:
5232:
5224:
5205:
5201:
5196:
5192:
5184:
5180:
5173:
5166:
5148:
5140:
5136:
5112:
5104:
5100:
5075:
5071:
5064:
5040:
5036:
5031:
5009:
5005:
5001:
4996:
4992:
4988:
4982:
4976:
4973:
4968:
4964:
4955:
4934:
4926:
4911:
4904:
4901:
4883:
4877:
4870:
4853:
4849:
4841:
4822:
4819:
4816:
4810:
4787:
4781:
4755:
4746:
4741:
4727:
4707:
4687:
4664:
4658:
4649:
4643:
4617:
4590:
4580:
4560:
4556:
4553:
4533:
4513:
4505:
4501:
4496:
4483:
4472:
4468:
4464:
4459:
4455:
4448:
4440:
4436:
4432:
4427:
4423:
4416:
4408:
4404:
4397:
4389:
4385:
4378:
4370:
4366:
4359:
4351:
4347:
4337:
4328:
4319:
4317:
4301:
4281:
4269:
4267:
4248:
4242:
4236:
4233:
4224:
4220:
4215:
4211:
4207:
4201:
4198:
4195:
4189:
4181:
4177:
4164:
4156:
4152:
4141:
4137:
4130:
4125:
4120:
4117:
4114:
4110:
4106:
4098:
4094:
4090:
4087:
4084:
4079:
4075:
4071:
4066:
4062:
4055:
4047:
4031:
4027:
4006:
3984:
3980:
3966:
3964:
3945:
3939:
3933:
3930:
3921:
3917:
3912:
3908:
3904:
3898:
3895:
3892:
3886:
3878:
3858:
3847:
3834:
3828:
3822:
3811:
3807:
3804:
3800:
3796:
3793:
3790:
3787:
3784:
3776:
3763:
3740:
3734:
3714:
3689:
3664:
3655:
3642:
3634:
3631:
3623:
3614:
3599:
3595:
3592:
3588:
3581:
3578:
3575:
3569:
3564:
3560:
3556:
3553:
3550:
3542:
3538:
3525:
3519:
3513:
3502:
3498:
3495:
3491:
3487:
3479:
3475:
3471:
3468:
3465:
3460:
3456:
3449:
3441:
3427:
3424:
3416:
3385:
3381:
3378:
3349:
3345:
3342:
3339:
3334:
3330:
3323:
3314:
3300:
3298:
3284:
3274:
3256:
3252:
3241:
3237:
3230:
3225:
3220:
3217:
3214:
3210:
3206:
3198:
3194:
3190:
3187:
3184:
3179:
3175:
3168:
3160:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3113:
3110:
3107:
3097:
3093:
3075:
3073:
3060:
3051:
3045:
3039:
3036:
3027:
3023:
3018:
3014:
3010:
3004:
3001:
2998:
2992:
2984:
2981:
2968:
2960:
2956:
2945:
2941:
2932:
2928:
2920:
2916:
2912:
2907:
2903:
2898:
2894:
2886:
2882:
2873:
2869:
2861:
2853:
2848:
2844:
2839:
2835:
2827:
2823:
2819:
2816:
2813:
2808:
2804:
2795:
2792:
2789:
2785:
2776:
2760:
2756:
2733:
2729:
2706:
2692:
2685:
2683:
2670:
2661:
2655:
2649:
2646:
2637:
2633:
2628:
2624:
2620:
2614:
2611:
2608:
2602:
2594:
2590:
2577:
2566:
2563:
2560:
2554:
2550:
2536:
2530:
2526:
2517:
2513:
2507:
2504:
2501:
2496:
2493:
2490:
2486:
2474:
2468:
2464:
2455:
2451:
2447:
2439:
2435:
2431:
2428:
2425:
2420:
2416:
2407:
2404:
2401:
2397:
2388:
2366:
2360:
2356:
2349:
2341:
2338:
2335:
2329:
2325:
2316:
2295:
2289:
2285:
2281:
2278:
2275:
2267:
2261:
2257:
2248:
2242:
2238:
2217:
2211:
2208:
2205:
2197:
2189:
2187:
2185:
2180:
2167:
2159:
2155:
2144:
2140:
2133:
2128:
2123:
2120:
2117:
2113:
2109:
2101:
2097:
2093:
2090:
2087:
2082:
2078:
2071:
2068:
2059:
2050:
2036:
2033:
2023:
2019:
1991:
1987:
1964:
1960:
1946:
1944:
1925:
1919:
1911:
1907:
1903:
1895:
1883:
1879:
1870:
1866:
1860:
1857:
1854:
1848:
1840:
1826:
1822:
1818:
1815:
1812:
1804:
1803:learning rate
1785:
1782:
1779:
1773:
1770:
1761:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1714:
1706:
1702:
1698:
1693:
1689:
1682:
1679:
1671:
1667:
1658:
1654:
1650:
1642:
1638:
1629:
1626:
1623:
1619:
1610:
1596:
1576:
1550:
1544:
1533:
1521:
1497:
1491:
1465:
1459:
1451:
1447:
1443:
1437:
1431:
1424:are sampled,
1411:
1408:
1405:
1385:
1373:
1371:
1352:
1346:
1338:
1334:
1330:
1327:
1324:
1315:
1311:
1305:
1302:
1299:
1293:
1285:
1266:
1262:
1253:
1249:
1240:
1239:learning rate
1221:
1218:
1215:
1209:
1206:
1197:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1148:
1143:
1139:
1127:
1121:
1115:
1112:
1109:
1105:
1098:
1094:
1090:
1084:
1076:
1072:
1063:
1059:
1052:
1049:
1046:
1040:
1032:
1028:
1019:
1016:
1013:
1009:
1000:
986:
983:
980:
960:
948:
945:
943:
930:
921:
915:
907:
903:
899:
896:
893:
884:
880:
874:
871:
868:
862:
854:
851:
838:
835:
832:
829:
826:
823:
820:
817:
814:
811:
802:
797:
793:
781:
775:
769:
766:
763:
759:
752:
749:
743:
735:
731:
722:
719:
716:
712:
703:
687:
684:
681:
678:
674:
653:
627:
621:
615:
589:
583:
577:
555:
552:
549:
546:
542:
529:
527:
524:
511:
503:
499:
492:
487:
482:
479:
476:
472:
468:
460:
456:
452:
449:
446:
441:
437:
430:
427:
418:
409:
395:
370:
366:
359:
356:
348:
344:
337:
334:
326:
322:
318:
313:
309:
302:
290:
288:
274:
254:
234:
214:
191:
185:
173:
171:
167:
165:
157:
153:
149:
145:
141:
137:
133:
129:
125:
122:
118:
114:
110:
107:
105:
101:
97:
93:
89:
85:
80:
77:
73:
72:
67:
63:
62:
54:
50:
46:
42:
38:
34:
30:
26:
21:
6672:
6666:
6640:
6633:
6598:
6592:
6551:
6547:
6537:
6520:11572/196442
6503:
6497:
6472:
6468:
6462:
6427:
6421:
6386:
6380:
6355:
6351:
6345:
6310:
6304:
6278:
6272:
6239:
6235:
6229:
6199:(1): 32–54.
6196:
6192:
6186:
6151:
6145:
6124:cite journal
6103:
6082:cite journal
6061:
6044:
6040:
6034:
6007:
6003:
5997:
5970:
5964:
5939:
5935:
5925:
5916:
5906:
5887:
5881:
5860:cite journal
5839:
5829:
5822:
5785:
5779:
5760:
5754:
5737:
5733:
5727:
5706:cite journal
5691:
5650:
5646:
5636:
5617:
5611:
5592:
5586:
5567:
5561:
5535:
5529:
5355:
5351:
5343:
5334:
5330:
5326:
5322:
5282:
5222:
5164:
4953:
4924:
4902:
4899:
4868:
4839:
4744:
4581:
4503:
4497:
4320:
4273:
4182:
4178:
4048:
3970:
3879:
3848:
3777:
3656:
3543:
3539:
3442:
3304:
3275:
3161:
3079:
2985:
2982:
2777:
2693:
2689:
2595:
2591:
2389:
2230:, such that
2193:
2181:
2051:
1950:
1841:
1762:
1611:
1377:
1286:
1198:
1001:
952:
855:
852:
704:
533:
525:
410:
294:
177:
168:
160:
155:
151:
147:
143:
139:
135:
131:
127:
123:
120:
116:
112:
108:
91:
87:
81:
70:
69:
65:
60:
59:
58:
52:
48:
44:
40:
36:
32:
28:
24:
6047:(1): 5–20.
6723:Categories
6682:1807.11669
6475:: 96–121.
5521:References
5319:assignment
666:elements,
423:Univariate
6568:1530-9304
6489:0020-0255
6372:0020-0255
6256:1000-9000
6213:1089-778X
6110:CiteSeerX
6068:CiteSeerX
6012:CiteSeerX
5846:CiteSeerX
5790:CiteSeerX
5765:CiteSeerX
5667:1063-6560
5416:α
5412:∘
5267:τ
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