Differential Evolution Hossein Talebi Hassan Nikoo 1

Outline  History  Introduction  Differences of DE with other Eas  Difference vector  Mutation  Cross over  Selection  General DE  Parameter control  Variation of DE  Application  References  Hassan’s parts 2 Differential Evolution

history  Ken Price's attempts to solve the Chebychev Polynomial fitting Problem that had been posed to him by Rainer Storn. 3 Differential Evolution

Introduction  The original DE was developed for continuous value problems  Individuals are vectors  Distance and direction information from current population is used to guide the search process 4 Differential Evolution

Difference of DE with other EAs 1. mutation is applied first to generate trial vectors, then cross over is applied to produce offspring 2. mutation step size are not sampled from prior know PDF, it influenced by difference between individual of the current population 5 Differential Evolution

Difference Vector  Positions of individuals provide valuable information about fitness landscape.  At first, individuals are distributed and over the time they converge to a same solution  Differences largein beginning of evolution bigger step size (exploring)  Differences are small at the end of search process smaller step size (exploiting) 6 Differential Evolution

DE operators  Mutation  Crossover  Selection Differential Evolution 7

mutation  Mutation produces a trial vector for each individual  This trial vector then will be used by crossover operator to produce offspring  For each parent, we make a trial vector as follow: 8 Differential Evolution

mutation (cont) Where: Target vector Weighted Differential 9 Differential Evolution

Geometrical Illustration (mutation) 10 Differential Evolution

Crossover  DE crossover is a recombination of trial vector,,and parent vector, to produce offspring, : 11 Differential Evolution

Methods to determine  Binomial crossover: Problem dimention 12 Differential Evolution

Methods to determine  Exponential crossover: 13 Differential Evolution

Geometrical Illustration (crossover) 14 Differential Evolution

Selection  selecting an individual to take part in mutation to make the trial vector. Random selection  select a target vector. Random or Best individual  selection between parent and offspring to spring. Better survive 15 Differential Evolution

General DE Algorithm 16 Differential Evolution

Control Parameters Scaling factor  The smaller the value of the smaller the step size  small enough to allow differentials to exploit tight valleys, and large enough to maintain diversity.  Empirical results suggest that generally provides good performance 17 Differential Evolution

Control Parameters Recombination probability  The higher the more variation is introduced in the new population  Increasing often results in faster convergence, while decreasing increases search robustness 18 Differential Evolution

Variation of DE 1. Target vector is selection (x) 2. Number of difference vectors used (y) 3. How crossover points are determined (z) 19 Differential Evolution

 Target vector is the best individual in current population,  One differential vector is used.  Any of the crossover methods. 20 Differential Evolution

 Any method for Target vector selection  more than one difference vector  Any of the crossover methods  the larger the value of, the more directions can be explored per generation. 21 Differential Evolution

 is randomly selected  The closer is to 1, the more greedy the search process  Value of close to 0 favors exploration. 22 Differential Evolution

 At list two difference vectors. 1. calculated from the best vector and the parent vector 2. while the rest of the difference vectors are calculated using randomly selected vectors  Empirical studies have shown DE/current- to-best/2/bin shows good convergence characteristics 23 Differential Evolution

application  Multiprocessor synthesis  Neural network learning  Synthesis of modulators  Heat transfer parameter estimation  Radio network design  … 24 Differential Evolution

References 1. Computational Intelligence, an introduction,2 nd edition, Andries Engelbercht, Wiley 2. Differential Evolution - A simple and efficient adaptive scheme for global optimization over continuous spaces, Rainer Storn,Kenneth Price, Differential Evolution, homepage Differential Evolution 25

Thanks For Your Attention Any Question? Differential Evolution 26