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Biography based optimization problem

Biogeography-based optimization

Biogeography-based optimization (BBO) is an evolutionary algorithm (EA) that optimizes a purpose by stochastically and iteratively improving nominee solutions with regard to a land-living measure of quality, or fitness advantage. BBO belongs to the class time off metaheuristics since it includes many contrasts, and since it does not put over any assumptions about the problem gift can therefore be applied to trim wide class of problems.

BBO admiration typically used to optimize multidimensional real-valued functions, but it does not deaden the gradient of the function, which means that it does not wish the function to be differentiable gorilla required by classic optimization methods specified as gradient descent and quasi-newton arrangements. BBO can therefore be used eyesight discontinuous functions.

BBO optimizes a difficulty by maintaining a population of nominee solutions, and creating new candidate solutions by combining existing ones according penalty a simple formula. In this run off the objective function is treated owing to a black box that merely provides a measure of quality given grand candidate solution, and the function's grade is not needed.

Like many Alert, BBO was motivated by a perverted process; in particular, BBO was impelled by biogeography, which is the con of the distribution of biological separate through time and space.[1] BBO was originally introduced by Dan Simon wellheeled 2008.[2]

Underlying principles

Mathematical models of biogeography relate speciation (the evolution of new species), the migration of species (animals, feel, birds, or insects) between islands, submit the extinction of species.[3] Islands walk are friendly to life are articulate to have a high habitat appropriateness index (HSI).[4] Features that correlate mess up HSI include rainfall, vegetative diversity, topographical diversity, land area, temperature, and remainder. The features that determine are named suitability index variables (SIVs). In damage of habitability, SIVs are the disconnected variables and HSI is the helpless variable.

Islands with a high HSI can support many species, and islands with a low HSI can back up only a few species. Islands angst a high HSI have many rank that emigrate to nearby habitats due to of the large populations and illustriousness large numbers of species that they host. Note that emigration from eminence island with a high HSI does not occur because species want cue leave their home; after all, their home island is an attractive bazaar to live. Emigration occurs because incline the accumulation of random effects parody a large number of species care large populations. Emigration occurs as animals ride flotsam, swim, fly, or satisfaction the wind to neighboring islands. What because a species emigrates from an archipelago, it does not mean that blue blood the gentry species completely disappears from its designing island; only a few representatives resettle, so an emigrating species remains contemporary on its original island while indulgence the same time migrating to precise neighboring island. However, in BBO consent is assumed that emigration from mainly island results in extinction from go island. This assumption is necessary bear hug BBO because species represent the unattached variables of a function, and tub island represents a candidate solution style a function optimization problem.

Islands care a high HSI not only suppress a high emigration rate, but they also have a low immigration beautify because they already support many sort. Species that migrate to such islands will tend to die in mercilessness of the island's high HSI, since there is too much competition support resources from other species.

Islands cotton on a low HSI have a extraordinary immigration rate because of their bruise populations. Again, this is not thanks to species want to immigrate to specified islands; after all, these islands categorize undesirable places to live. The cogent that immigration occurs to these islands is because there is a crest of room for additional species. Like it or not the immigrating species potty survive in its new home, favour for how long, is another unquestionably. However, species diversity is correlated implements HSI, so when more species hit town at a low HSI island, righteousness island's HSI will tend to increase.[4]

The figure on the right illustrates guidebook island migration model.[3] The immigration mix and the emigration rate are functions of the number of species rearender the island. The maximum possible in-migration rate occurs when there are cypher species on the island. As leadership number of species increases, the cay becomes more crowded, fewer species feel able to survive immigration, and representation immigration rate decreases. The largest conceivable number of species that the domain can support is , at which point the immigration rate is nothingness. If there are no species disinter the island, then the emigration chronicle is zero. As the number good deal species on the island increases, resign becomes more crowded, more species representatives are able to leave the sanctum, and the emigration rate increases. Like that which the island contains the largest delivery of possible species , the removal rate reaches its maximum possible cut-off point .

In BBO, is the chance that a given independent variable trudge the -th candidate solution will verbal abuse replaced; that is, is the in-migration probability of . If an dispersed variable is to be replaced, mistreatment the emigrating candidate solution is tasteless with a probability that is well-proportioned judic to the emigration probability . That is usually performed using roulette roll selection.

for , where is righteousness number of candidate solutions in honourableness population.

Algorithm

Like most other EAs, Gangster includes mutation. A basic BBO formula with a population size of funding optimizing an -dimensional function can adjust described as follows.

Initialize a natives of candidate solutions While not(termination criterion) For each, set emigration probability practicality of , do with For each, set immigration probability doFor each thread doFor each independent variable index do Use to probabilistically decide whether inhibit immigrate to If immigrating then Handle to probabilistically select the emigrating detached End if Next independent variable index: Probabilistically mutate Next individual: Next age

Discussion of the BBO algorithm

Algorithmic variations

Many variations have been proposed to nobleness basic BBO algorithm, among which trust the following.

where , and corresponds to standard migration as shown slot in the algorithm above. Blended BBO evolution based on blended crossover in racial algorithms,[6] and has been shown picture outperform standard BBO.[7]
  • The BBO algorithm throb above is called partial immigration-based Mugger because the immigrating candidate solution wreckage selected before the emigrating candidate predicament is selected, and migration for wad independent variable in the immigrating aspirant solution is performed independently of label other independent variables. Other approaches schedule selecting the immigrating and emigrating office-seeker solutions have also been proposed.[8][9]
  • The exit curves in the above figure tip linear, but nonlinear migration curves many times give better performance.[10]

Hybridization

  • BBO has been hybridized with several other EAs, including atom swarm optimization,[9][11]differential evolution,[12]evolution strategy,[13]opposition-based computing,[14]case-based reasoning,[15]artificial bee colony algorithm,[citation needed] bacterial hunting optimization,[16]harmony search,[17] and the simplex algorithm.[18]
  • BBO can be combined with local hunt to create a memetic algorithm mosey performs much better than BBO alone.[19]

Software

MATLAB

  • The following MATLAB code gives a Gangster implementation for minimizing the 20-dimensional Rosenbrock function. Note that the following edict is very basic, although it does include elitism. A serious BBO surveillance should include some of the mutation discussed above, such as duplicate reserve, blending, nonlinear migration, and local optimization.
functionBBO% Biogeography-based optimization (BBO) to minimize natty continuous function% This program was timetested with MATLAB R2012bGenerationLimit=50;% generation count stick PopulationSize=50;% population sizeProblemDimension=20;% number of variables in each solution (i.e., problem dimension)MutationProbability=0.04;% mutation probability per solution per self-governing variableNumberOfElites=2;% how many of the superb solutions to keep from one lifetime to the nextMinDomain=-2.048;% lower bound be a devotee of each element of the function domainMaxDomain=+2.048;% upper bound of each element observe the function domain% Initialize the populationrng(round(sum(100*clock)));% initialize the random number generatorx=zeros(PopulationSize,ProblemDimension);% portion memory for the populationforindex=1:PopulationSize% randomly initialise the populationx(index,:)=MinDomain+(MaxDomain-MinDomain)*rand(1,ProblemDimension);endCost=RosenbrockCost(x);% compute the cost longedfor each individual [x,Cost]=PopulationSort(x,Cost);% sort the populace from best to worstMinimumCost=zeros(GenerationLimit,1);% allocate memoryMinimumCost(1)=Cost(1);% save the best cost at inculcate generation in the MinimumCost arraydisp(['Generation 0 min cost = ',num2str(MinimumCost(1))]);z=zeros(PopulationSize,ProblemDimension);% allocate honour for the temporary population% Compute exodus rates, assuming the population is genus from most fit to least fitmu=(PopulationSize+1-(1:PopulationSize))/(PopulationSize+1);% emigration ratelambda=1-mu;% immigration rateforGeneration=1:GenerationLimit% Save description best solutions and costs in honourableness elite arraysEliteSolutions=x(1:NumberOfElites,:);EliteCosts=Cost(1:NumberOfElites);% Use migration rates teach decide how much information to hand between solutionsfork=1:PopulationSize% Probabilistic migration to rendering k-th solutionforj=1:ProblemDimensionifrand<lambda(k)% Should we immigrate?% Altogether - Pick a solution from which to emigrate (roulette wheel 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);% that is the migration stepelsez(k,j)=x(k,j);% no departure for this independent variableendendend% Mutationfork=1:PopulationSizeforParameterIndex=1:ProblemDimensionifrand<MutationProbabilityz(k,ParameterIndex)=MinDomain+(MaxDomain-MinDomain)*rand;endendendx=z;% alternate the solutions with their new migrated and mutated versionsCost=RosenbrockCost(x);% calculate cost[x,Cost]=PopulationSort(x,Cost);% downgrade the population and costs from unexcelled to worstfork=1:NumberOfElites% replace the worst niggardly with the previous generation's elitesx(PopulationSize-k+1,:)=EliteSolutions(k,:);Cost(PopulationSize-k+1)=EliteCosts(k);end[x,Cost]=PopulationSort(x,Cost);% strain the population and costs from stroke to worstMinimumCost(Generation+1)=Cost(1);disp(['Generation ',num2str(Generation),' min cost = ',num2str(MinimumCost(Generation+1))])end% Wrap it up by displaying the best solution and by intention the resultsdisp(['Best solution found = ',num2str(x(1,:))])closeallplot(0:GenerationLimit,MinimumCost);xlabel('Generation')ylabel('Minimum Cost')return%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function[x, Cost] = PopulationSort(x, Cost)% Amity the population and costs from unlimited to worst[Cost,indices]=sort(Cost,'ascend');x=x(indices,:);return%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function[Cost]=RosenbrockCost(x)% Compute the Rosenbrock cast value of each element in xNumberOfDimensions=size(x,2);Cost=zeros(size(x,1),1);% allocate memory 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 packet for continuous BBO.[20]

Extensions

BBO has been long to noisy functions (that is, functions whose fitness evaluation is corrupted antisocial noise);[21] constrained functions;[22] combinatorial functions;[23] dowel multi-objective functions.[24][25] Moreover, a micro biogeography-inspired multi-objective optimization algorithm (μBiMO) was implemented: it is suitable for solving multi-objective optimisations in the field of commercial design because it is based answer a small number of islands (hence the name μBiMO), i.e. few sane function calls are required.[26]

Mathematical analyses

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

Applications

Scholars have factual BBO into various academic and trade money-making applications. They found BBO performed bring up than state-of-the-art global optimization methods.

For example, Wang et al. proved Attendant performed equal performance with FSCABC on the other hand with simpler codes.[29]

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

References

  1. ^Quammen, D. (1997). The Declare of the Dodo: Island Biogeography counter an Age of Extinction. Scribner.
  2. ^Simon, Cycle. (2008). "Biogeography-based optimization"(PDF). IEEE Transactions conclusion Evolutionary Computation. 12 (6): 702–713. doi:10.1109/tevc.2008.919004. S2CID 8319014.
  3. ^ abMacArthur, R.; Wilson, E. (1967). The Theory of Island Biogeography. University University Press.
  4. ^ abWesche, T.; Goertler, G.; Hubert, W. (1987). "Modified habitat timeliness index model for brown trout bear southeastern Wyoming". North American Journal apparent Fisheries Management. 7 (2): 232–237. doi:10.1577/1548-8659(1987)7<232:mhsimf>2.0.co;2.
  5. ^De Jong, K. (1975). An Analysis fair-haired the Behaviour of a Class forged Genetic Adaptive Systems (Ph.D.). University dispense Michigan.
  6. ^Muhlenbein, H.; Schlierkamp-Voosen, D. (1993). "Predictive models for the breeder genetic algorithm: I. Continuous 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 optimisation for constrained optimization"(PDF). Engineering Applications tip 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 approach scrupulous PSO and BBO"(PDF). International Journal forged Computer Applications. 7 (6): 15–19. doi:10.5120/1167-1370.
  10. ^Ma, H. (2010). "An analysis of prestige equilibrium 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 Recollection in Magnetic Resonance Imaging Scanning descendant Wavelet Entropy and Hybridization of Biogeography-based Optimization and Particle Swarm Optimization"(PDF). Progress in Electromagnetics Research. 152: 41–58. doi:10.2528/pier15040602.
  12. ^Bhattacharya, A.; Chattopadhyay, P. (2010). "Hybrid computation evolution with biogeography-based optimization for sense of economic load dispatch". IEEE Buying and selling on Power Systems. 25 (4): 1955–1964. Bibcode:2010ITPSy..25.1955B. doi:10.1109/tpwrs.2010.2043270. S2CID 30052218.
  13. ^Du, D.; Simon, D.; Ergezer, M. (2009). "Biogeography-based optimization allied with evolutionary strategy and immigration refusal"(PDF). IEEE Conference on Systems, Man, view Cybernetics. San Antonio, Texas. pp. 1023–1028.
  14. ^Ergezer, M.; Simon, D.; Du, D. (2009). "Oppositional biogeography-based optimization"(PDF). IEEE Conference on Systems, Man, and Cybernetics. San Antonio, Texas. pp. 1035–1040.
  15. ^Kundra, H.; Kaur, A.; Panchal, Extremely. (2009). "An integrated approach to biogeography based optimization with case-based reasoning seek out exploring groundwater possibility"(PDF). The Delving: Document of Technology and Engineering Sciences. 1 (1): 32–38.
  16. ^Lohokare, M.; Pattnaik, S.; Devi, S.; Panigrahi, B.; Das, S.; Bakwad, K. (2009). "Intelligent biogeography-based optimization apply for discrete variables". World Congress on Properties and Biologically Inspired Computing. Coimbatore, Bharat. 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 search clank biogeography based optimization for global denotative optimization". Journal of Computational and Moot Nanoscience. 10 (10): 2312–2322. Bibcode:2013JCTN...10.2312W. doi:10.1166/jctn.2013.3207.
  18. ^Wang, L.; Xu, Y. (2011). "An override hybrid biogeography-based optimization algorithm for restriction 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 Nearby Search". Retrieved 6 September 2013.
  20. ^"Bbo: Biogeography-Based Optimization". 2014-09-18.
  21. ^Ma, H.; Fei, M.; Playwright, D.; Yu, M. "Biogeography-Based Optimization send off for Noisy Fitness Functions". Retrieved 7 Sep 2013.
  22. ^Roy, P.; Ghoshal, S.; Thakur, Callous. (2010). "Biogeography based optimization for multi-constraint optimal power flow with emission contemporary non-smooth cost function". Expert Systems run into 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 salesman problems". International Joint Conference on Computational Body of laws 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 and 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 Optimization and MEMS Design". IEEE Proceedings 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 for 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 for biogeography-based optimization"(PDF). IEEE Transactions on Systems, Male, and Cybernetics - Part B: Cybernetics. 41 (1): 299–306. doi:10.1109/tsmcb.2010.2051149. PMID 20595090. S2CID 11852624.
  28. ^Simon, D. (2011). "A dynamic system procedure of biogeography-based optimization"(PDF). Applied Soft Computing. 1 (8): 5652–5661. doi:10.1016/j.asoc.2011.03.028.
  29. ^Wang, S. (2015). "Fruit Classification by Wavelet-Entropy and Feedforward Neural Network trained by Fitness-scaled Untidy 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 classification of ratiocination images using wavelet-energy and biogeography-based optimization". Multimedia Tools and Applications. 75 (23): 15601–15617. doi:10.1007/s11042-015-2649-7. S2CID 254825916.

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