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

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

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 = and 7/16 = 00011

Gray Coding ● Gray Encoding – Increase of 1 step changes only 1 bit. – Example: ● ● ● ● ● – Convert Bin-Gray:Gray-Bin:

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)

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

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

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

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}

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)]

A Real Coded Genetic Algorithm Problems with real crossover

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.

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

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

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

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

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

Experimental Results

Experimental Results Summary ● 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.

Experimental Results Summary ● 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.

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

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.