D IFFERENTIAL E VOLUTION By Fakhroddin Noorbehbahani EA course, Dr. Mirzaee December,

A GENDA Preface Basic Differential Evolution Difference Vectors Mutation Crossover Selection General Differential Evolution Algorithm Control Parameters Geometrical Illustration DE/ x/y/z 2

A GENDA Variations to Basic Differential Evolution Hybrid Differential Evolution Strategies Population-Based Differential Evolution Self-Adaptive Differential Evolution Differential Evolution for Discrete-Valued Problems Constraint Handling Approaches Comparison with other algorithms Applications 3

P REFACE Price and Storn in 1995 Chebychev Polynomial fitting Problem 3rd place at the First International Contest on evolutionary Computation (1stICEO) 1996, the best genetic type of algorithm for solving the real-valued test function suite. stochastic, population-based search strategy Main characteristics Guide search with distance and direction information from the current population original DE strategies for continuous-valued landscapes 4

B ASIC D IFFERENTIAL E VOLUTION mutation is applied first to generate a trial vector, which is then used within the crossover operator to produce one offspring, mutation step sizes are not sampled from a prior known probability distribution function. mutation step sizes are influenced by differences between individuals of the current population 5

D IFFERENCE V ECTORS Position of individuals and fitness Over time, as the search progresses, the distances between individuals become smaller The magnitude of the initial distances between individuals is influenced by the size of the population Distances between individuals are a very good indication of the diversity of the current population Use difference vector to determine the step size total number of differential perturbations n v is the number of differentials used n s is the population size 6

M UTATION 7

C ROSSOVER Binomial crossover Exponential crossover 8

S ELECTION Random Selection To select the individuals from which difference vectors are calculated. The target vector is either randomly selected or the best individual is selected Deterministic Selection To construct the population for the next generation, the offspring replaces the parent if the fitness of the offspring is better than its parent; otherwise the parent survives to the next generation. This ensures that the average fitness of the population does not deteriorate. 9

G ENERAL D IFFERENTIAL E VOLUTION A LGORITHM 10

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C ONTROL P ARAMETERS population size, n s scale factor, β probability of recombination, Pr 12

P OPULATION S IZE The size of the population has a direct influence on the exploration ability of DE algorithms. The more individuals there are in the population, the more differential vectors are available, and the more directions can be explored The computational complexity per generation increases with the size of the population. Empirical studies provide the guideline that n s 10n x 13

S CALE F ACTOR The scaling factor, β (0,), controls the amplification of the differential variations, (x i 2 xi 3 ). The smaller the value of β, the smaller the mutation step sizes Smaller step sizes can be used to explore local areas. slower convergence Larger values for β facilitate exploration, but may cause the algorithm to overshoot optima As the population size increases, the scaling factor should decrease. 14

R ECOMBINATION P ROBABILITY This parameter controls the number of elements of the parent, x i (t), that will change. The higher the probability of recombination, the more variation is introduced in the new population, thereby increasing diversity and increasing exploration. Increasing p r often results in faster convergence, while decreasing pr increases search robustness 15

G EOMETRICAL I LLUSTRATION 16

G EOMETRICAL I LLUSTRATION 17

G EOMETRICAL I LLUSTRATION 18

E XAMPLE :P EAK FUNCTION 19

G EOMETRICAL I LLUSTRATION Generation 1: DEs population and difference vector distributions 20

G EOMETRICAL I LLUSTRATION Generation 6: The population coalesces around the two main minima 21

G EOMETRICAL I LLUSTRATION Generation 12: The difference vector distribution contains three main clouds – one for local searches and two for moving between the two main minima. 22

G EOMETRICAL I LLUSTRATION Generation 16: The population is concentrated on the main minimum 23

G EOMETRICAL I LLUSTRATION Generation 20: Convergence is imminent. The difference vectors automatically shorten for a fine-grained, local search. 24

G EOMETRICAL I LLUSTRATION Generation 26: The population has almost converged. 25

G EOMETRICAL I LLUSTRATION Generation 34: DE finds the global minimum. 26

DE/ X / Y / Z DE/best/1/ z DE/ x/n v /z DE/rand-to-best/ n v /z 27

DE/ X / Y / Z DE/current-to-best/1+ nv/z DE/rand/1/bin vs. DE/current-to-best/2/bin DE/rand/1/bin maintains good diversity DE/current-to-best/2/bin shows good convergence characteristics Dynamically switch between these two strategies 28

V ARIATIONS TO B ASIC D IFFERENTIAL E VOLUTION Hybrid Differential Evolution Strategies Gradient-Based Hybrid Differential Evolution Acceleration operator : to improve convergence speed Migration operator : to improve ability for escaping local optima Acceleration operator uses gradient descent to adjust the best individual toward obtaining a better position if the mutation and crossover operators failed to improve x( t), replaces the worst individual in the new population, C(t+1). 29

M IGRATION OPERATOR Gradient decent speed up but local minima Migration operator increase population diversity Generate new individual from best individuals Applied when diversity is too small i.e.: 30

H YBRID D IFFERENTIAL E VOLUTION WITH A CCELERATION AND M IGRATION 31

E VOLUTIONARY A LGORITHM -B ASED H YBRIDS DE reproduction process as a crossover operator in a simple GA Rank-Based Crossover Operator for Differential Evolution To select individuals to calculate difference vectors x i1 (t) precedes x i2 (t) if f(x i1 (t)) > f(x i2 (t)). 32

R ANK -B ASED M UTATION O PERATOR FOR D IFFERENTIAL E VOLUTION 33

O THER V ARIATIONS TO B ASIC DE Population-Based Differential Evolution Improve exploration by using 2 population set Initialize with n s pairs Rejected individual by selection put in auxiliary pop Self-Adaptive Differential Evolution Dynamic Parameters Self-Adaptive Parameters 34

D IFFERENTIAL E VOLUTION FOR D ISCRETE - V ALUED P ROBLEMS Angle Modulated DE where x is a single element from a set of evenly separated intervals determined by the required number of bits that need to be generated 35

A NGLE M ODULATED D IFFERENTIAL E VOLUTION 36

B INARY D IFFERENTIAL E VOLUTION 37

C ONSTRAINT H ANDLING A PPROACHES Penalty methods Converting the constrained problem to an unconstrained problem By changing the selection operator of DE, infeasible solutions can be rejected 38

C OMPARISON WITH GA AND PSO 39

C OMPARISON 40

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A PPLICATIONS 1) General Optimization Framework "Mystic" by Mike McKerns, Caltech. 2) Multiprocessor synthesis. 3) Neural network learning. 4) Chrystallographic characterization. 5) Synthesis of modulators. 6) Heat transfer parameter estimation in a trickle bed reactor. 7) Scenario-Integrated Optimization of Dynamic Systems. 8) Optimal Design of Shell-and-Tube Heat Exchangers. 9) Optimization of an Alkylation Reaction. 10) Optimization of Thermal Cracker Operation. 11) Optimization of Non-Linear Chemical Processes. 12) Optimum planning of cropping patterns. 13) Optimization of Water Pumping System. 14) Optimal Design of Gas Transmission Network. 15) Differential Evolution for Multi-Objective Optimization 16) Physiochemistry of Carbon Materials. 17) Radio Network Design. 18) Reflectivity Curve Simulation. 45

C OMMERCIAL SOFT 1) Built in optimizer in MATHEMATICA's function Nminimize (since version 4.2). 2) MATLAB's GA toolbox contains a variant of DE. 3) Digital Filter Design. 4) Diffraction grating design. 5) Electricity market simulation. 6) Auto2Fit. 7) LMS Virtual Lab Optimization. 8) Optimization of optical systems. 9) Finite Element Design. 46

A PPLICATION : BUMP PROBLEM 47

A PPLICATION : BUMP PROBLEM 48

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R EFERENCES [1] [2] Andries P. Engelbrecht,(2007),Computational Intelligence: An Introduction, 2nd Edition., ISBN: Andries P. Engelbrecht [3] Price, K.; Storn, R.M.; Lampinen, J.A. (2005). Differential Evolution: A Practical Approach to Global Optimization. Springer. ISBN ence/foundations+of+computations/book/ Differential Evolution: A Practical Approach to Global OptimizationISBN ence/foundations+of+computations/book/ [4] Feoktistov, V. (2006). Differential Evolution: In Search of Solutions. Springer. ISBN Differential Evolution: In Search of SolutionsISBN [5] J. Vesterstrom and R. Thomson, A comparative study of differential evolution, particle swarm optimization, and evolutionary algorithms on numerical benchmark problems, Proc. of IEEE Congress on Evolutionary Computation, 2004, pp. 1980–

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T HANKS FOR YOUR ATTENTION 52