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

2. 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

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

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

5. 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.

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

7. 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

8. 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

9. 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

10. 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

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

12. 12 Benchmark Functions Step function: separable, irregular, unimodal

13. 13 Benchmark Functions Rastrigin: nonseparable, regular, multimodal

14. 14 Benchmark Functions Rosenbrock: nonseparable, regular, unimodal

15. 15 Benchmark Functions Schwefel 2.22: nonseparable, irregular, unimodal

16. 16 Benchmark Functions Schwefel 2.26: separable, irregular, multimodal

17. 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)

18. 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

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

20. 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

21. 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}

22. 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

23. 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.

24. 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