genetic algorithms for real parameter optimization written by alden h. wright department of computer...

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.