Genetic Algorithms for Real Parameter Optimization

1. Written by Alden H. Wright Department of Computer Science University of Montana Presented by Tony Morelli 11/01/2004 Genetic Algorithms for Real Parameter Optimization

2. Background • Usual method of applying GAs to real-parameter is to encode each parameter using binary coding or Gray coding • Parameters are concatenated together to create a chromosome. • Each Bit position corresponds to a gene • Each Bit value corresponds to an allele • This paper's approach • Chromosome – Vector of real parameters • Gene – A real number • Allele – A real value

3. Binary Coding • Binary Coding • Break valid range into segments and associate value based on segment • If 0 < x < 4 and 5 bits are used • 32 Segments • Each segment = 1/8 (.125) • 3/8 = 00011 and 7/16 = 00011

4. Gray Coding • Gray Encoding • Increase of 1 step changes only 1 bit. • Example: • 0 0000 • 1 0001 • 2 0011 • 3 0010 • 4 0110 • Convert Bin-Gray: Gray-Bin:

5. Crossover • One Point Binary Crossover • 1 Crossover point is selected • Bits from 1 parent and to the left of the crossover point are combined with bits from the other parent and to the right of the crossover point • Crossing over 5/32 (00101) and 27/32 (11011) between bits 3 and 4 yields 7/32 (00111) and 25/32 (11001)

6. Mutation • Binary Coded GA • Probability of mutation is low • If mutation occurs, bit changes from 1 to 0 or 0 to 1 • If change from 0 to 1 • Binary coding – Change is in the positive direction • Gray coding – Change in either direction • If change from 1 to 0 • Binary coding – Change is in the negative direction • Gray coding – Change is in either direction

7. Schemata • Similarity Template • Describes a subset of the space of chromosomes • {01*} = {010, 011} • Connected schemata are the most meaningful • They capture locality info about the function

8. A Real Coded Genetic Algorithm • Standard GA • 1. A method for choosing the initial population • 2. A Scaling function that creates a nonnegative fitness function • 3. Find the sampling rate of an individual • 4. Pick which individuals are allowed to reproduce • 5. Reproduction operators to produce new individuals • 6. A method for choosing which reproduction operator to apply

9. A Real Coded Genetic Algorithm • Standard GA, only steps 5 and 6 require bitwise manipulation. • Real crossover is almost the same is in binary • Take the list of real numbers from one parent, combine them with a list from the other parent • {5,6,7,8},{1,2,3,4} combine at crossover point between 2 and 3 to create children {5,6,3,4} and {1,2,7,8}

10. A Real Coded Genetic Algorithm • Real Mutation • Mutation is performed if chromosome is selected • Direction is then chosen (50/50 either positive or negative • Amount of mutation is determined • Original parameter is x, range [a,b], mutation size M • Direction is positive • Mutated parameter is uniformly chosen from [x,min(M,b)]

11. A Real Coded Genetic Algorithm • Problems with real crossover

12. A Real Coded Genetic Algorithm • Linear Crossover • From 2 parent points, 3 new points are generated: • (1/2)p1 + (1/2)p2, (3/2)p1 - (1/2)p2, (-1/2)p1+(3/2)p2 • (1/2)p1 + (1/2)p2 is the midpoint of p1 and p2 • The others are on the line determined by p1 and p2 • The best 2 of the 3 points are sent to the next generation • Disadvantage - Highly disrupted of schemata and is not compatible with the schema theorem described in the next slide.

13. Schemata Analysis for Real-Allele Genetic Algorithms • Restrict some or all of the parameters to subintervals of their possible ranges • Ii denotes the interval • If parameter space is [-1,1]x[-1,1]x[-1,1] and schema is [-1,1]x[0,1]x[-1,0], the m-tuple would be *I2I3 • The probability of an individual being selected for reproduction is the ration of its fitness to the average fitness of the entire population

14. Schemata Analysis for Real-Allele Genetic Algorithms • The expected proportion of individuals of schema s that are selected after reproduction is shown by:

15. Experimental Results • Tested on DeJongs 5 problems, 2 other problems (Schaffer, Caruana, Eshelman, and Das)

16. Experimental Results • 2 Point Crossover • The Elitist Strategy • Bakers Selection Procedure • Population size of 20 • Crossover rate of 0.8 • Gray coding was used for Binary-Coded • GA was run for 1000 trials, except for one of the cases which was run for 5000 • Best Performance – Min value over all trials • Best Offline Performance – Average over function evaluations of the best value obtained up to that function evaluation

17. Experimental Results • Multiple runs were done to tune parameters • Binary • 1000 experiments at each mutation rate from 0.005 to 0.05 in steps of 0.005 • Real • 1000 experiments were done at each combination of a mutation size and mutation rate • Mutation sizes 0.1-0.3 steps of 0.1 • Mutation rate from 0.05 to 0.3 steps of 0.05 • 50% real crossover and 50% linear crossover

18. Experimental Results

19. Experimental ResultsSummary • Real Coded algorithm with 50% real crossover and 50% linear crossover performed better than 100% real crossover on all problems • Real-Coded with both types of crossover performed better than the binary-coded on 7/9 problems • Real-Coded algorithm with real crossover performed better than the binary-coded on 5/9 problems.

20. Experimental ResultsSummary • On problems F4 and F7 the mixed crossover real-coded did much better than the other 2 • On F5 (Shekel's Foxholes) the binary coded algorithm did much better than the real-coded algorithms. • This problem is well suited for binary GAs • When it was rotated 30 degrees, the difference between the real-coded and the binary was less, but the binary GA still outperformed.

21. Conclusions • Results showed that the real-coded GA based on a mixture of real and linear crossover gave superior results to binary-coded GAs on most test problems • Real-Coded GA with both linear and real crossover outperformed the GA with only 1 of them. • Strengths of a real-coded GA • 1. Increased Effeciency (No need to convert bit strings) • 2. Increased Precision (Using real numbers) • 3. Can use different mutation and crossover techniques

22. Questions/Comments • Word of the day: ENVISAGE • en·vis·age • 1. To conceive an image or a picture of, especially as a future possibility: envisaged a world at peace. • 2. To consider or regard in a certain way.