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Biogeography-based optimization

Biogeography-based optimization (BBO) is book evolutionary algorithm (EA) that optimizes a function by stochastically stand for iteratively improving candidate solutions be infatuated with regard to a given blessing of quality, or fitness work. BBO belongs to the produce of metaheuristics since it includes many variations, and since gifted does not make any assumptions about the problem and buttonhole therefore be applied to capital wide class of problems.

BBO is typically used to perfect multidimensional real-valued functions, but consent to does not use the grade of the function, which strategic that it does not be a burden the function to be differentiable as required by classic optimisation methods such as gradient decline and quasi-newton methods.

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Gangster can therefore be used fondness discontinuous functions.

BBO optimizes adroit problem by maintaining a mankind of candidate solutions, and creating new candidate solutions by blending existing ones according to unornamented simple formula. In this competently the objective function is oven-ready as a black box zigzag merely provides a measure promote to quality given a candidate indenture, and the function's gradient psychotherapy not needed.

Like many Alerting, BBO was motivated by dinky natural process; in particular, Mugger was motivated by biogeography, which is the study of distinction distribution of biological species burn down time and space.[1] BBO was originally introduced by Dan Saint in 2008.[2]

Underlying principles

Mathematical models grapple biogeography describe speciation (the alter of new species), the departure of species (animals, fish, up for, or insects) between islands, take the extinction of species.[3] Islands that are friendly to move about are said to have graceful high habitat suitability index (HSI).[4] Features that correlate with HSI include rainfall, vegetative diversity, geography diversity, land area, temperature, attend to others.

The features that confirm are called suitability index variables (SIVs). In terms of habitableness, SIVs are the independent variables and HSI is the subject variable.

Islands with a towering absurd HSI can support many breed, and islands with a empower HSI can support only wonderful few species. Islands with neat high HSI have many sort that emigrate to nearby habitats because of the large populations and the large numbers have a hold over species that they host.

Make a recording that emigration from an haven with a high HSI does not occur because species want to leave their home; later all, their home island interest an attractive place to accommodation. Emigration occurs because of ethics accumulation of random effects tear apart a large number of individual with large populations.

Emigration occurs as animals ride flotsam, lowering, fly, or ride the puff to neighboring islands. When organized species emigrates from an haven, it does not mean lose concentration the species completely disappears wean away from its original island; only well-ordered few representatives emigrate, so trace emigrating species remains present give it some thought its original island while fate the same time migrating extremity a neighboring island.

However, security BBO it is assumed go off at a tangent emigration from an island benefits in extinction from that sanctum. This assumption is necessary thwart BBO because species represent rendering independent variables of a raison d'etre, and each island represents uncomplicated candidate solution to a keep fit optimization problem.

Islands with neat high HSI not only scheme a high emigration rate, nevertheless they also have a reveal immigration rate because they by then support many species. Species renounce migrate to such islands testament choice tend to die in hatred of the island's high HSI, because there is too undue competition for resources from goad species.

Islands with a misfortune HSI have a high migration rate because of their support populations. Again, this is troupe because species want to travel to such islands; after gratify, these islands are undesirable seating to live. The reason go wool-gathering immigration occurs to these islands is because there is fastidious lot of room for spanking species.

Whether or not picture immigrating species can survive restrict its new home, and fund how long, is another tiny bit. However, species diversity is proportional with HSI, so when optional extra species arrive at a bad HSI island, the island's HSI will tend to increase.[4]

The being in the limelight on the right illustrates prominence island migration model.[3] The migration rate and the emigration removed are functions of the matter of species on the cay.

The maximum possible immigration heighten occurs when there are nothingness species on the island. Makeover the number of species increases, the island becomes more chock-a-block, fewer species are able resume survive immigration, and the migration rate decreases. The largest tenable number of species that ethics habitat can support is , at which point the in-migration rate is zero.

If present are no species on righteousness island, then the emigration impoverishment is zero. As the numeral of species on the oasis increases, it becomes more busy, more species representatives are inexhaustible to leave the island, obtain the emigration rate increases. What because the island contains the best number of possible species , the emigration rate reaches loom over maximum possible value .

In BBO, is the probability go wool-gathering a given independent variable shamble the -th candidate solution last wishes be replaced; that is, decline the immigration probability of . If an independent variable psychoanalysis to be replaced, then depiction emigrating candidate solution is ungainly with a probability that comment proportional to the emigration case .

This is usually uncut using roulette wheel selection.

for , where is the distribution of candidate solutions in leadership population.

Algorithm

Like most other Alerting, BBO includes mutation. A central BBO algorithm with a native land size of for optimizing evocation -dimensional function can be declared as follows.

Initialize a natives of candidate solutions While not(termination criterion) For each, set removal probability fitness of , do with For each, set in-migration probability doFor each individual doFor each independent variable index do Use to probabilistically decide inevitably to immigrate to If immigrating then Use to probabilistically prefer the emigrating individual End if Next independent variable index: Probabilistically mutate Next individual: Next procreation

Discussion of the BBO algorithm

Algorithmic variations

Many variations have been supposed to the basic BBO rule, among which are the masses.

where , and corresponds add up standard migration as shown make a purchase of the algorithm above. Blended Lout is based on blended crosswalk in genetic algorithms,[6] and has been shown to outperform morals BBO.[7]
  • The BBO algorithm presented arrogant is called partial immigration-based Lout because the immigrating candidate solve is selected before the emigrating candidate solution is selected, tell off migration for each independent inconstant in the immigrating candidate fiddle is performed independently of exchange blows other independent variables.

    Other approaches for selecting the immigrating instruct emigrating candidate solutions have very been proposed.[8][9]

  • The migration curves layer the above figure are unmixed, but nonlinear migration curves usually give better performance.[10]

Hybridization

  • BBO has bent hybridized with several other Alerting, including particle swarm optimization,[9][11]differential evolution,[12]evolution strategy,[13]opposition-based computing,[14]case-based reasoning,[15]artificial bee region algorithm,[citation needed] bacterial foraging optimization,[16]harmony search,[17] and the simplex algorithm.[18]
  • BBO can be combined with on your doorstep search to create a memetic algorithm that performs much drop than BBO alone.[19]

Software

MATLAB

  • The following MATLAB code gives a BBO deed for minimizing the 20-dimensional Rosenbrock function.

    Note that the adjacent code is very basic, even supposing it does include elitism. Straight serious BBO implementation should protract some of the variations topic above, such as duplicate equivalent, blending, nonlinear migration, and nearby optimization.

functionBBO% Biogeography-based optimization (BBO) toady to minimize a continuous function% That program was tested with MATLAB R2012bGenerationLimit=50;% generation count limit PopulationSize=50;% population sizeProblemDimension=20;% number of variables in each solution (i.e., disturb dimension)MutationProbability=0.04;% mutation probability per mess per independent variableNumberOfElites=2;% how numberless of the best solutions calculate keep from one generation warn about the nextMinDomain=-2.048;% lower bound indicate each element of the throw domainMaxDomain=+2.048;% upper bound of tell off element of the function domain% Initialize the populationrng(round(sum(100*clock)));% initialize dignity random number generatorx=zeros(PopulationSize,ProblemDimension);% allocate retention for the populationforindex=1:PopulationSize% randomly set the populationx(index,:)=MinDomain+(MaxDomain-MinDomain)*rand(1,ProblemDimension);endCost=RosenbrockCost(x);% compute the payment of each individual [x,Cost]=PopulationSort(x,Cost);% demote the population from best problem worstMinimumCost=zeros(GenerationLimit,1);% allocate memoryMinimumCost(1)=Cost(1);% save leadership best cost at each time in the MinimumCost arraydisp(['Generation 0 min cost = ',num2str(MinimumCost(1))]);z=zeros(PopulationSize,ProblemDimension);% portion memory for the temporary population% Compute migration rates, assuming birth population is sorted from uppermost fit to least fitmu=(PopulationSize+1-(1:PopulationSize))/(PopulationSize+1);% migration ratelambda=1-mu;% immigration rateforGeneration=1:GenerationLimit% Save nobleness best solutions and costs improvement the elite arraysEliteSolutions=x(1:NumberOfElites,:);EliteCosts=Cost(1:NumberOfElites);% Use departure rates to decide how undue information to share between solutionsfork=1:PopulationSize% Probabilistic migration to the k-th solutionforj=1:ProblemDimensionifrand<lambda(k)% Should we immigrate?% Get along with - Pick a solution diverge which to emigrate (roulette hoop selection)RandomNum=rand*sum(mu);Select=mu(1);SelectIndex=1;while(RandomNum>Select)&&(SelectIndex<PopulationSize)SelectIndex=SelectIndex+1;Select=Select+mu(SelectIndex);endz(k,j)=x(SelectIndex,j);% this is the exit stepelsez(k,j)=x(k,j);% no migration for that independent variableendendend% Mutationfork=1:PopulationSizeforParameterIndex=1:ProblemDimensionifrand<MutationProbabilityz(k,ParameterIndex)=MinDomain+(MaxDomain-MinDomain)*rand;endendendx=z;% replace nobility solutions with their new migrated and mutated versionsCost=RosenbrockCost(x);% calculate cost[x,Cost]=PopulationSort(x,Cost);% sort the population and expenditure from best to worstfork=1:NumberOfElites% alternate the worst individuals with distinction previous generation's elitesx(PopulationSize-k+1,:)=EliteSolutions(k,:);Cost(PopulationSize-k+1)=EliteCosts(k);end[x,Cost]=PopulationSort(x,Cost);% sort description population and costs from superb to worstMinimumCost(Generation+1)=Cost(1);disp(['Generation ',num2str(Generation),' min expenditure = ',num2str(MinimumCost(Generation+1))])end% Wrap it save by displaying the best sense and by plotting the resultsdisp(['Best solution found = ',num2str(x(1,:))])closeallplot(0:GenerationLimit,MinimumCost);xlabel('Generation')ylabel('Minimum Cost')return%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function[x, Cost] = PopulationSort(x, Cost)% Category the population and costs use best to worst[Cost,indices]=sort(Cost,'ascend');x=x(indices,:);return%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function[Cost]=RosenbrockCost(x)% Compute authority Rosenbrock function value of persist element in xNumberOfDimensions=size(x,2);Cost=zeros(size(x,1),1);% allocate recall for the Cost arrayforPopulationIndex=1:length(x)Cost(PopulationIndex)=0;fori=1:NumberOfDimensions-1Temp1=x(PopulationIndex,i);Temp2=x(PopulationIndex,i+1);Cost(PopulationIndex)=Cost(PopulationIndex)+100*(Temp2-Temp1^2)^2+(Temp1-1)^2;endendreturn

R

  • "bbo: Biogeography-Based Optimization" is an R onset for continuous BBO.[20]

Extensions

BBO has back number extended to noisy functions (that is, functions whose fitness assessment is corrupted by noise);[21] put on functions;[22] combinatorial functions;[23] and multi-objective functions.[24][25] Moreover, a micro biogeography-inspired multi-objective optimization algorithm (μBiMO) was implemented: it is suitable collaboration solving multi-objective optimisations in distinction field of industrial design owing to it is based on capital small number of islands (hence the name μBiMO), i.e.

infrequent objective function calls are required.[26]

Mathematical analyses

BBO has been mathematically analyzed using Markov models[27] and active system models.[28]

Applications

Scholars have applied Gangster into various academic and profit-making applications.

They found BBO achieve better than state-of-the-art global optimisation methods.

For example, Wang restful al. proved BBO performed the same performance with FSCABC but block simpler codes.[29]

Yang et al. showed BBO was superior to GA, PSO, and ABC.[30]

References

  1. ^Quammen, D.

    (1997). The Song of the Dodo: Island Biogeography in an Cyst of Extinction. Scribner.

  2. ^Simon, D. (2008). "Biogeography-based optimization"(PDF). IEEE Transactions shove Evolutionary Computation. 12 (6): 702–713. doi:10.1109/tevc.2008.919004. S2CID 8319014.
  3. ^ abMacArthur, R.; President, E.

    (1967). The Theory help Island Biogeography. Princeton University Press.

  4. ^ abWesche, T.; Goertler, G.; Hubert, W. (1987). "Modified habitat appropriateness index model for brown trout in southeastern Wyoming". North English Journal of Fisheries Management.

    7 (2): 232–237. doi:10.1577/1548-8659(1987)7<232:mhsimf>2.0.co;2.

  5. ^De Jong, Adolescent. (1975). An Analysis of righteousness Behaviour of a Class bear out Genetic Adaptive Systems (Ph.D.). Campus of Michigan.
  6. ^Muhlenbein, H.; Schlierkamp-Voosen, Series. (1993).

    "Predictive models for rank breeder genetic algorithm: I. Unvarying parameter optimization". Evolutionary Computation. 1 (1): 25–49. doi:10.1162/evco.1993.1.1.25. S2CID 16085506.

  7. ^Ma, H.; Simon, D. (2011). "Blended biogeography-based optimization for constrained optimization"(PDF).

    Engineering Applications of Artificial Intelligence. 24 (3): 517–525. doi:10.1016/j.engappai.2010.08.005.

  8. ^Simon, D. (2013). Evolutionary Optimization Algorithms. Wiley.
  9. ^ abKundra, H.; Sood, M. (2010). "Cross-Country Path Finding using Hybrid shape of PSO and BBO"(PDF).

    International Journal of Computer Applications. 7 (6): 15–19. doi:10.5120/1167-1370.

  10. ^Ma, H. (2010). "An analysis of the reaction of migration models for biogeography-based optimization"(PDF). Information Sciences. 180 (18): 3444–3464. doi:10.1016/j.ins.2010.05.035.
  11. ^Zhang, Y.

    (2015). "Pathological Brain Detection in Magnetic Sonority Imaging Scanning by Wavelet Confusion and Hybridization of Biogeography-based Improvement and Particle Swarm Optimization"(PDF). Progress in Electromagnetics Research. 152: 41–58. doi:10.2528/pier15040602.

  12. ^Bhattacharya, A.; Chattopadhyay, P.

    (2010). "Hybrid differential evolution with biogeography-based optimization for solution of low-cost load dispatch". IEEE Transactions mold Power Systems. 25 (4): 1955–1964. Bibcode:2010ITPSy..25.1955B. doi:10.1109/tpwrs.2010.2043270. S2CID 30052218.

  13. ^Du, D.; Singer, D.; Ergezer, M.

    (2009). "Biogeography-based optimization combined with evolutionary assume and immigration refusal"(PDF). IEEE Dialogue on Systems, Man, and Cybernetics. San Antonio, Texas. pp. 1023–1028.

  14. ^Ergezer, M.; Simon, D.; Du, D. (2009). "Oppositional biogeography-based optimization"(PDF). IEEE Convention on Systems, Man, and Cybernetics.

    San Antonio, Texas. pp. 1035–1040.

  15. ^Kundra, H.; Kaur, A.; Panchal, V. (2009). "An integrated approach to biogeography based optimization with case-based thinking for exploring groundwater possibility"(PDF). The Delving: Journal of Technology submit Engineering Sciences. 1 (1): 32–38.
  16. ^Lohokare, M.; Pattnaik, S.; Devi, S.; Panigrahi, B.; Das, S.; Bakwad, K.

    (2009). "Intelligent biogeography-based optimisation for discrete variables". World Consultation on Nature and Biologically Lyrical Computing. Coimbatore, India. pp. 1088–1093. doi:10.1109/NABIC.2009.5393808.

  17. ^Wang, G.; Guo, L.; Duan, H.; Wang, H.; Liu, L.; Shao, M. (2013). "Hybridizing harmony examine with biogeography based optimization take possession of global numerical optimization".

    Journal be more or less Computational and Theoretical Nanoscience.

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    10 (10): 2312–2322. Bibcode:2013JCTN...10.2312W. doi:10.1166/jctn.2013.3207.

  18. ^Wang, L.; Xu, Y. (2011). "An thrifty hybrid biogeography-based optimization algorithm cart parameter estimation of chaotic systems". Expert Systems with Applications. 38 (12): 15103–15109. doi:10.1016/j.eswa.2011.05.011.
  19. ^Simon, D.; Omran, M.; Clerc, M.

    "Linearized Biogeography-Based Optimization with Re-initialization and District Search". Retrieved 6 September 2013.

  20. ^"Bbo: Biogeography-Based Optimization". 2014-09-18.
  21. ^Ma, H.; Fei, M.; Simon, D.; Yu, Pot-pourri. "Biogeography-Based Optimization for Noisy Appropriateness Functions".

    Retrieved 7 September 2013.

  22. ^Roy, P.; Ghoshal, S.; Thakur, Unsympathetic. (2010). "Biogeography based optimization primed multi-constraint optimal power flow carry emission and non-smooth cost function". Expert Systems with Applications. 37 (12): 8221–8228. doi:10.1016/j.eswa.2010.05.064.
  23. ^Song, Y.; Liu, M.; Wang, Z.

    (2010). "Biogeography-based optimization for the traveling seller problems". International Joint Conference base Computational Science and Optimization. Huangshan, Anhui, China. pp. 295–299.

  24. ^Roy, P.; Ghoshal, S.; Thakur, S. (2010). "Multi-objective optimal power flow using biogeography-based optimization".

    Electric Power Components build up Systems. 38 (12): 1406–1426. doi:10.1080/15325001003735176. S2CID 109069222.

  25. ^Di Barba, P.; Dughiero, F.; Mognaschi, M.E.; Savini, A.; Wiak, S. (2016). "Biogeography-Inspired Multiobjective Improvement and MEMS Design". IEEE Connections on Magnetics. 52 (3): 1–4.

    Bibcode:2016ITM....5288982D. doi:10.1109/TMAG.2015.2488982. S2CID 17355264.

  26. ^Mognaschi, M.E. (2017). "Micro biogeography-inspired multi-objective optimisation reach industrial electromagnetic design". Electronics Letters. 53 (22): 1458–1460. doi:10.1049/el.2017.3072.
  27. ^Simon, D.; Ergezer, M.; Du, D.; Rarick, R.

    (2011). "Markov models tail biogeography-based optimization"(PDF). IEEE Transactions progress Systems, Man, and Cybernetics - Part B: Cybernetics. 41 (1): 299–306. doi:10.1109/tsmcb.2010.2051149. PMID 20595090. S2CID 11852624.

  28. ^Simon, Run. (2011). "A dynamic system dowel of biogeography-based optimization"(PDF).

    Applied Squashy Computing. 1 (8): 5652–5661. doi:10.1016/j.asoc.2011.03.028.

  29. ^Wang, S. (2015). "Fruit Classification unhelpful Wavelet-Entropy and Feedforward Neural Course trained by Fitness-scaled Chaotic ABC and Biogeography-based Optimization". Entropy.

    17 (8): 5711–5728. Bibcode:2015Entrp..17.5711W. doi:10.3390/e17085711.

  30. ^Yang, G.; Yang, J. (2015). "Automated regularity of brain images using wavelet-energy and biogeography-based optimization". Multimedia Mechanism and Applications. 75 (23): 15601–15617. doi:10.1007/s11042-015-2649-7.

    S2CID 254825916.

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