Genetic Algorithms for Real Parameter Optimization

Written by Alden H. WrightDepartment of Computer Science

University of Montana

Presented by Tony Morelli11/01/2004

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 = 00011 and 7/16 = 00011

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:

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 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

Experimental Results

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.

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.

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.