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Derivative-Free Optimization: Biogeography-Based Optimization Dan Simon Cleveland State University 1

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2 Outline 1.Biogeography 2.Biogeography-Based Optimization 3.Benchmark Functions and Results 4.Sensor Selection: A Real-World Problem 5.BBO Code Walk-Through

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3 Biogeography The study of the geographic distribution of biological organisms Mauritius 1600s

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4 Biogeography Species migrate between “islands” via flotsam, wind, flying, swimming, …

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5 Biogeography Habitat Suitability Index (HSI): Some islands are more suitable for habitation than others Suitability Index Variables (SIVs): Habitability is related to features such as rainfall, topography, diversity of vegetation, temperature, etc.

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6 Biogeography As habitat suitability improves: – The species count increases – Emigration increases (more species leave the habitat) – Immigration decreases (fewer species enter the habitat)

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7 Biogeography-Based Optimization 1.Initialize a set of solutions to a problem 2.Compute “fitness” (HSI) for each solution 3.Compute S,, and for each solution 4.Modify habitats (migration) based on, 5.Mutatation 6.Typically we implement elitism 7.Go to step 2 for the next iteration if needed

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Biogeography-Based Optimization = the probability that the immigrating individual’s solution feature is replaced = the probability that an emigrating individual’s solution feature migrates to the immigrating individual ---- immigrating island (individual) emigrating islands (individuals) 8

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9 Benchmark Functions 14 standard benchmark functions were used to evaluate BBO relative to other optimizers. Ackley Fletcher-Powell Griewank Penalty Function #1 Penalty Function #2 Quartic Rastrigin Rosenbrock Schwefel 1.2 Schwefel 2.21 Schwefel 2.22 Schwefel 2.26 Sphere Step

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10 Benchmark Functions Functions can be categorized as Separable or nonseparable – for example, (x+y) vs. xy Regular or irregular – for example, sin x vs. abs(x) Unimodal or multimodal – for example, x 2 vs. cos x

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11 Benchmark Functions Penalty function #1: nonseparable, regular, unimodal

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12 Benchmark Functions Step function: separable, irregular, unimodal

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13 Benchmark Functions Rastrigin: nonseparable, regular, multimodal

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14 Benchmark Functions Rosenbrock: nonseparable, regular, unimodal

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15 Benchmark Functions Schwefel 2.22: nonseparable, irregular, unimodal

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16 Benchmark Functions Schwefel 2.26: separable, irregular, multimodal

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17 Optimization Algorithms Ant colony optimization (ACO) Biogeography-based optimization (BBO) Differential evolution (DE) Evolutionary strategy (ES) Genetic algorithm (GA) Population-based incremental learning (PBIL) Particle swarm optimization (PSO) Stud genetic algorithm (SGA)

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18 Average performance of 100 simulations (n = 50) ACOBBODEESGAPBILPSOSGA Ackley182100146197 232192103 Fletcher1013100385494415917799114 Griewank16211727269651628311023100 Penalty 12.2E71.2E49.7E41.3E62.5E52.8E72.1E6100 Penalty 25.0E571558624.2E41.1E45.4E56.4E4100 Quartic32132621176700828504.8E48570100 Rastrigin454100397536421634470134 Rosenbrock17111022537164281861516100 Schwefel 1.2202100391425166606592110 Schwefel 2.21161100227162184265179146 Schwefel 2.226881002901094500861665142 Schwefel 2.26108118137140142177142100 Sphere134710025091090627851000109 Step24811230281355132711161100

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19 Aircraft Engine Sensor Selection Health estimation Better maintenance Better control performance

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20 Aircraft Engine Sensor Selection What sensors should we use? Measure pressures, temperatures, speeds 11 sensors; some can be duplicated Estimate efficiencies and airflow capacities Optimize estimation accuracy and cost Use a Kalman filter for health estimation

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21 Aircraft Engine Sensor Selection Suppose we want to pick N objects out of K classes while choosing from each class no more than M times. Example: We have red balls, blue balls, and green balls (K=3). We want to pick 4 balls (N=4) with each color chosen no more than twice (M=2). 6 Possibilities: {B, B, G, G}, {R, B, G, G}, {R, B, B, G}, {R, R, G, G}, {R, R, B, G}, {R, R, B, B}

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22 Aircraft Engine Sensor Selection Pick N objects out of K classes while choosing from each class no more than M times. q(x)= (1 + x + x 2 + … + x M ) K = 1 + q 1 x + q 2 x 2 + … + q N x N + … + x MK Multinomial theorem: The number of unique combinations (order independent) is q N

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23 Aircraft Engine Sensor Selection Example: Pick 20 objects out of 11 classes while choosing from each class no more than 4 times. q(x) = (1 + x + x 2 + x 3 + x 4 ) 11 = 1 + … + 3,755,070 x 20 + …+ x 44 21 hours of CPU time for an exhaustive search. We need a quick suboptimal search strategy.

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24 Aircraft Engine Sensor Selection Average and best performance of 100 Monte Carlo simulations. Computational savings = 99.99% (21 hours 8 seconds). ACOBBODEESGAPBILPSOSGA Mean8.228.018.068.158.048.188.148.02 Best8.127.197.608.058.028.808.068.02 BBO.m

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