Differential Evolution

1. Differential Evolution HosseinTalebi Hassan Nikoo

2. 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

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

4. 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

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

6. 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 large in beginning of evolution bigger step size (exploring) • Differences are small at the end of search process smaller step size (exploiting)

7. DE operators • Mutation • Crossover • Selection

8. 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:

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

10. Geometrical Illustration (mutation)

11. Crossover • DE crossover is a recombination of trial vector, ,and parent vector , to produce offspring, :

12. Methods to determine • Binomial crossover: Problem dimention

13. Methods to determine • Exponential crossover:

14. Geometrical Illustration (crossover)

15. 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

16. General DE Algorithm

17. Control Parameters Scalingfactor • 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

18. 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

19. Variation of DE • Target vector is selection (x) • Number of difference vectors used (y) • How crossover points are determined (z)

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

21. 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.

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

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

24. application • Multiprocessor synthesis • Neural network learning • Synthesis of modulators • Heat transfer parameter estimation • Radio network design • …

25. References • Computational Intelligence, an introduction,2nd edition, AndriesEngelbercht, Wiley • Differential Evolution - A simple and efficient adaptive scheme for global optimization over continuous spaces, Rainer Storn,Kenneth Price,1995 • Differential Evolution, homepagehttp://www.icsi.berkeley.edu/~storn/code.html

26. Thanks For Your Attention Any Question?