1. Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello Department of Mathematics and Computer Science University of Catania

2. Selection and Recombination - Lecture 42 Selection and Recombination  1. Review of the previous lecture (a) How a simple evolutionary algorithm works (b) Some crossover and mutation operators  2. Selection and reproduction  3. Recombination  4. A sample algorithm

3. Selection and Recombination - Lecture 43 Selection and Reproduction  Selection is not normally regarded as a search operator although it does in search significantly.  Selection can be used either before or after search operators.  When selection is used before search operators, the process of choosing the next generation from the union of all parents and offspring is sometimes called reproduction.  The generational gap of an EA refers to the overlap (i.e., individuals that did not go through any search operators) between the old and new generations.  The two extremes are generational EAs and steady-state EAs.  Elitism = copying the best individual to the next generation.

4. Selection and Recombination - Lecture 44 Fitness Proportional Selection  Also known as roulette wheel selection.  Use raw fitness in computing selection probabilities. Does not allow negative fitness values.  Weakness: Domination of super individuals in early generations and slow convergence in later generations.  Fitness scaling has often been used in early days to combat the two problems.

5. Selection and Recombination - Lecture 45 Fitness Scaling  Simple scaling The ith individual's fitness is defined as: f scaled (t) = f original (t) - f worst (t); where t is the generation number and f worst (t) the fitness of the worst individual so far.  Sigma scaling The ith individual's fitness is defined as: where c is a constant, e.g., 2, f * (t) is the average fitness in the current population, and  f (t) is the standard deviation of the fitness in the current population.

6. Selection and Recombination - Lecture 46  Power scaling The ith individual's fitness is defined as: f scaled (t) = (f original (t)) k where k > 0.  Exponential scaling The ith individual's fitness is defined as: f scaled (t) = exp( f original (t)/T ) where T > 0 is the temperature, approaching zero.

7. Selection and Recombination - Lecture 47 Ranking  Linear ranking Assume that the population size is , rank 0 indicates the worst individual and rank  -1 the best. The probability of selecting the ith individual is: Where  (  ) indicates the expected number of offspring produced by the worst (best) individuals. Note that: Hence  +  = 2. So 1    2 and  = 2- .  Power ranking C below is a normalising factor. 0<  < .

8. Selection and Recombination - Lecture 48  Geometric ranking :  Exponential ranking:  Xin Yao ranking:

9. Selection and Recombination - Lecture 49 Recombination for Real-valued Representation  Discrete Recombination does not change actual (gene) values. Very similar to the crossover operators on binary strings.  Intermediate Recombination does change actual (gene) values. Usually based on some kind of average/mixture among multiple parents.

10. Selection and Recombination - Lecture 410 Discrete Recombination  Multi-point Recombination Similar to that for the binary representation.  Global Discrete Recombination Similar to uniform crossover for the binary representation.  Geometric explanation.

11. Selection and Recombination - Lecture 411 Intermediate Recombination  With two parents Given x 1 and x 2 :  With more parents Given x 1 and x 2 : where

12. Selection and Recombination - Lecture 412 Other Recombination  Heuristic Recombination Assume x 2 is no worse than x 1 : where u is a uniformly distributed random number in [0,1].  Simplex Recombination Randomly select a group (> 2) of parents. Assume x 1 is the best individual and x 2 the worst in the group. Compute the centroid, c, of the group without x 2.  Geometric Recombination Can be generalised to multiple parents.

13. Selection and Recombination - Lecture 413 Quadratic Recombination  Let x ij be the j-th component of the vectors x i,  i  1,2,3 , j  1,…,n , where n is the dimensionality. We approximate the position of P 4 using the quadratic interpolation method as follow.  One offspring is generated from three parents.

14. Selection and Recombination - Lecture 414 What Does Quadratic Recombination Mean Note that we are minimising “fitness” here

15. Selection and Recombination - Lecture 415 A Hybrid EA With Local Search  1. Initialize  individuals at random.  2. Perform local search from each individual.  3. REPEAT (a) Generate 3 points P 1, P 2, P 3 by global discrete recombination. (b) Perform a quadratic approximation using P 1, P 2, P 3 to produce a point P 4. (c) Perform a local search from P 4 and update P 4 with the search result. (Sounds Lamarkian.) (d) Place P 1, P 2, P 3, P 4 into the population and do a (  + 4) truncation selection.  4. UNTIL termination criteria are met.

16. Selection and Recombination - Lecture 416 Local Search With Random Memorising  Store best solutions in a memory.  Retrieve a random one (old best) when a new best solution is found.  Search along the direction of old  new, i.e., the direction of:

17. Selection and Recombination - Lecture 417 Experimental Studies  18 multimodal benchmark functions.  Population size 30  Maximum function evaluation 500,000.  50 independent runs for each function.

18. Selection and Recombination - Lecture 418 Benchmark Functions (f 8 - f 25 )

19. Selection and Recombination - Lecture 419 Results on f 8 -f 13

20. Selection and Recombination - Lecture 420 Results on f 8 -f 13 with population size 50 and 60

21. Selection and Recombination - Lecture 421 Results on f 14 -f 25

22. Selection and Recombination - Lecture 422 Results on f 16 -f 23 with population size 10

23. Selection and Recombination - Lecture 423 Summary  Different problems require different search operators and selection schemes. There is no universally best one.  Using real vectors is usually more appropriate than binary strings for function optimisation.  Many search operators are heuristics-based. Domain knowledge can often be incorporated into search operators and representation.

24. Selection and Recombination - Lecture 424 References  T. Bäck, D. B. Fogel, and Z. Michalewicz (eds.), Handbook of Evolutionary Computation, IOP Publ. Co. & Oxford University Press, 1997. Part C. (in the School library)  K.-H. Liang, X. Yao and C. S. Newton, “Combining landscape approximation and local search in global optimization" Proc. of the 1999 Congress on Evolutionary Computation, Vol. 2, IEEE Press, Piscataway, NJ, USA, pp.1514-1520, July 1999.