Post on 11-Jan-2016
Kuban State UniversityKrasnodar, Russia
Genetic algorithms: an Genetic algorithms: an introductionintroduction
Artem Eremin,
j. researcher, IMMI KSU
Genetic Algorithms: an introduction
Motivation
Genetic Algorithms: an introduction
Motivation
Genetic Algorithms: an introduction
Motivation
Material properties (Cij) - ???
• Doppler laservibrometry for measuring
out-of-plane velocities
experimental data
• Time-of-Flight (TOF) with wavelet
transform
TOF
Wavelet transform
g
dc
TOF
min ?ijC
F 2, ,
1
: ( )pN
g j g jj
F c c
Genetic Algorithms: an introduction
Optimization
“Optimization is the process of making something better”
Every day we subconsciouslysolve some optimization problems!
Genetic Algorithms: an introduction
Optimization
Genetic Algorithms: an introduction
Minimum-seeking algorithms
1. Exhaustive Search = Brute Force
2. Analytical Optimization
3. Nelder-Mead downhill Simplex Method
4. Optimization based on Line Minimization (the coordinate search method, the steepest descent algorithm, Newton’s method, Davidon-Fletcher-Powell (DFP) algorithm, etc …)
2
( ) 0, ?f
fx
x
Genetic Algorithms: an introduction
Minimum-seeking algorithms
1 – 4 can converge to a local minimum!
Natural optimization methodsNot the panacea, but …
Genetic algorithms(Holland, 1975)
Simulated annealing(Kirkpatrick et al., 1983)
Particle swarm optimization(Parsopoulos and Vrahatis, 2002)
Evolutionary algorithms(Schwefel, 1995)
No derivatives, large search spaces, “nature-based”
Genetic Algorithms: an introduction
Biological background (Cell and Chromosomes)• Every animal cell is a complex of many small “factories” working
together; the center of this all is the cell nucleus; the nucleus contains the genetic information in chromosomes - strings of DNA
• Each chromosome contains a set of genes - blocks of DNA
• Each gene determines some aspect of the organism (like eye colour)
• A collection of genes is sometimes called a genotype
• A collection of aspects (like eye colour) is sometimes called a phenotype
Genetic Algorithms: an introduction
Biological background (Reproduction)Organisms produce a number of offspring similar to themselves but can have variations due to:
– Mutations (random changes)
– Sexual reproduction (offspring have combinations of features inherited from each parent)
+
Genetic Algorithms: an introduction
Biological background (Natural Selection)
• The Origin of Species: “Preservation of favourable variations and rejection of unfavourable variations.”
• There are more individuals born than can survive, so there is a continuous struggle for life.
• Individuals with an advantage have a greater chance for survive: survival of the fittest.
• Important aspects in natural selection are: adaptation to the environment and isolation of populations in different groups which cannot mutually mate
Genetic Algorithms: an introduction
Genetic algorithms (GA)
GA were initially developed by John Holland, University of Michigan (1970’s)
Popularized by his student David Goldberg (solved some very complex engineering problems, 1989)
Based on ideas from Darwinian Evolution
Provide efficient techniques for optimization and machine learning applications; widely used in business, science and engineering
Genetic Algorithms: an introduction
GA main features
• Optimizes with continuous or discrete variables
• Doesn’t require derivative information
• Simultaneously searches from a wide sampling of the cost surface
• Deals with a large number of variables
• Is well suited for parallel computers
• Optimizes variables with extremely complex cost surfaces (they can jump out of a local minimum)
• Provides a list of optimum variables, not just a single solution
• May encode the variables so that the optimization is done with the encoded variables
• Works with numerically generated data, experimental data, or analytical functions.
Genetic Algorithms: an introduction
To start with…
Genotype space = {0,1}L
Phenotype space
Encoding (representation)
Decoding(inverse representation)
011101001
010001001
10010010
10010001
( , ) sin 4 1.1 sin 2
0 , 10
f x y y x x y
x y
x
y
0 10
10
Genetic Algorithms: an introduction
To start with…
Gene – a single encoding of part of the solution space, i.e. either single bits or short blocks of adjacent bits that encode an element of the candidate solution
1 0 0 1 4.1
xChromosome – a string of genes that represents a solution
0 1 1 0 1 0 0 1+
Population – the number of chromosomes available to test
Genetic Algorithms: an introduction
Chromosomes
Chromosomes can be:– Bit strings (0110, 0011, 1101, …)– Real numbers (33.2, -12.11, 5.32, …)– Permutations of elements (1234, 3241, 4312, …)– Lists of rules (R1, R2, R3, …Rn…)– Program elements(genetic programming)– …
Chromosome = array of Nvar variables (genes) pi
var1 2[ , , ... , ]Nchromosome p p p
( )cost f chromosome
chromosomespopN
Genetic Algorithms: an introduction
How does it works?
produce an initial population of individuals
evaluate the fitness of all individuals
while termination condition not met do
select fitter individuals for reproduction
recombine between individuals
mutate individuals
evaluate the fitness of the modified individuals
generate a new population
End while
So…
Genetic Algorithms: an introduction
How does it works?
The Evolutionary Cycle
selection
population evaluation
modification
discard
deleted members
parents
modifiedoffspring
evaluated offspring
initiate & evaluate
Or so…
Genetic Algorithms: an introduction
Generation of the initial population[ , ]i i ip a b
random[0,1] ( )
1,i i i i
var
p a b a
i N
Coding: 4.25 01101101...
s1 = 1111010101 f (s1) = 7s2 = 0111000101 f (s2) = 5s3 = 1110110101 f (s3) = 7s4 = 0100010011 f (s4) = 4s5 = 1110111101 f (s5) = 8s6 = 0100110000 f (s6) = 3
Ex.Npop=6
Genetic Algorithms: an introduction
Selection
[0.4,0.6]
keep rate pop
rate
N X N
X
or
, save!i tri f f
We are kind! Let’s save everybody!
Mating pool
Genetic Algorithms: an introduction
Selection
,keep pop keepN N N
1. Pairing from top to bottom1
2 1...
keep
keep
NN
2. Random pairing ,1
,2
random[1, ]random[1, ]
c keep
c keep
s Ns N
3. Weighted random pairingroulette wheel weighting
Genetic Algorithms: an introduction
Selection
The roulette wheel method:
21n
3
Area is proportional to fitness value
Individual i will have a probability to be chosen i
if
if
)(
)(
4
We repeat the extraction as many times as it is necessary
Genetic Algorithms: an introduction
Selection
4. Tournament selection
1. Zenit 50
2. CSKA 48
…
16. Krylia Sovietov 14
a) randomly pick a small subsetb) perform a “tournament”c) “the winner takes it all”
Tournament + Threshold = No SORTING!!!
Genetic Algorithms: an introduction
Mating (Crossover)
• Choose a random point on the two parents• Split parents at this crossover point• Create children by exchanging tails• Pc typically in range (0.6, 0.9)
Simple 1-point crossover
• Performance with 1 Point Crossover depends on the order that variables occur in the representation– more likely to keep together genes that are near
each other– Can never keep together genes from opposite ends
of string– This is known as Positional Bias– Can be exploited if we know about the structure of
our problem, but this is not usually the case
Genetic Algorithms: an introduction
Mating (Crossover)n-point crossover
• Choose n random crossover points• Split along those points• Glue parts, alternating between parents• Generalisation of 1 point (still some positional bias)
Genetic Algorithms: an introduction
Mating (Crossover)Uniform crossover
Uniform crossover looks at each bit in the parents and randomly assigns the bit from one parent to one offspring and the bit from the other parent to the other offspring
Genetic Algorithms: an introduction
Mutation
• Alter each gene (or, bit) independently with a probability pm
• pm is called the mutation rate
• Typically between 1/Npop and 1/[s]
Genetic Algorithms: an introduction
Crossover or/and Mutation
• A long debate: which one is better / necessary / main-background
• Answer (at least, rather wide agreement):
– it depends on the problem, but
– in general, it is good to have both
– both have another role
– mutation-only-GA is possible, crossover-only-GA would not work
Genetic Algorithms: an introduction
Crossover or/and Mutation
• Exploration: Discovering promising areas in the search space, i.e. gaining information on the problem
• Exploitation: Optimising within a promising area, i.e. using information
• There is co-operation AND competition between them
• Crossover is explorative, it makes a big jump to an area somewhere “in between” two (parent) areas
• Mutation is exploitative, it creates random small diversions, thereby staying near (in the area of ) the parent
• Only crossover can combine information from two parents
• Only mutation can introduce new information
• To hit the optimum you often need a ‘lucky’ mutation
Genetic Algorithms: an introduction
Mapping real values on bit strings
pi [ai, bi] R represented by {a1,…,aL} {0,1}L
• [ai, bi] {0,1}L must be invertible (one phenotype per
genotype)
: {0,1}L [ai, bi] defines the representation
• Only 2L values out of infinite are represented
• L determines possible maximum precision of solution
• High precision long chromosomes (slow evolution)
1
10
( ,..., ) ( 2 ) [ , ]2 1
Lji i
L i L j i iLj
b aa a a a a b
Real valued problems
Genetic Algorithms: an introduction
General scheme of floating point mutations
• Uniform mutation:
• Analogous to bit-flipping (binary) or random resetting
(integers)
1 2 1 2( , ,..., ) ' ( ' , ' ,..., ' )pop popN Ns p p p s p p p
, ' ,i i i ip p a b
' drawn randomly (uniform) from ,i i ip a b
Floating point mutations
Genetic Algorithms: an introduction
• Non-uniform mutations:– Many methods proposed,such as time-varying range of
change etc.– Most schemes are probabilistic but usually only make a
small change to value– Most common method is to add random deviate to each
variable separately, taken from N(0, ) Gaussian distribution and then curtail to range
– Standard deviation controls amount of change (2/3 of deviations will lie in range (- to + )
Floating point mutations
Genetic Algorithms: an introduction
• Discrete:– each gene value in offspring z comes from one of its
parents (x,y) with equal probability: zi = xi or yi
– Could use n-point or uniform• Intermediate
– exploits idea of creating children “between” parents (hence a.k.a. arithmetic recombination)
– zi = xi + (1 - ) yi where : 0 1.– The parameter can be:
• constant: uniform arithmetical crossover• variable (e.g. depend on the age of the
population) • picked at random every time
Crossover for real valued GAs
Genetic Algorithms: an introduction
• Parents: x1,…,xn and y1,…,yn• Pick a single gene (k) at random, • child1 is:
• reverse for other child. e.g. with = 0.5
nkkk xxyxx ..., ,)1( , ..., ,1
Single arithmetic crossover
Genetic Algorithms: an introduction
• Parents: x1,…,xn and y1,…,yn• Pick random gene (k) after this point mix values• child1 is:
• reverse for other child. e.g. with = 0.5
nx
kx
ky
kxx )1(
ny ..., ,
1)1(
1 , ..., ,
1
Simple arithmetic crossover
Genetic Algorithms: an introduction
• Most commonly used• Parents: x1,…,xn and y1,…,yn• child1 is:
• reverse for other child. e.g. with = 0.5
yaxa )1(
“Whole” arithmetic crossover
Genetic Algorithms: an introduction
First generation (random values)
Tournament selection
SBX crossover
Select fittest individual
Start new generation
Good results? Enough iterations?
resultYes
No
micro-GA
Genetic Algorithms: an introduction
Benefits of GA
• Concept is easy to understand
• Modular–separate from application (representation); building blocks can be used in hybrid applications
• Supports multi-objective optimization
• Good for “noisy”environment
• Always results in an answer, which becomes better and better with time
• Can easily run in parallel
• The fitness function can be changed from iteration to iteration, which allows incorporating new data in the model if it becomes available
Genetic Algorithms: an introduction
Issues with GA
Choosing parameters:–Population size–Crossover and mutation probabilities–Selection, deletion policies–Crossover, mutation operators, etc.–Termination criteria
Performance:–Can be too slow but covers a large search space–Is only as good as the fitness function
Examples
Genetic Algorithms: an introduction
Experimental specimens
4 CFRP–plates
2,35mmH [0 ,90 ,90 ,0 ]o o o o
2,25mmH [0 ,0 ,0 ,0 ]o o o o
60%fV 58%fV
Genetic Algorithms: an introduction
Material properties
107 2% GPa; 8.9 2% GPa
2.82 2% GPa; 4.38 1% GPa
0.25 0.32; 0.49 0.56
x y z
yz xz yz
xz xy yz
E E E
G G G
110.5 7.0 7.0 0 0 0
7.0 13.8 8.2 0 0 0
7.0 8.2 13.8 0 0 0GPa
0 0 0 2.8 0 0
0 0 0 0 4.37 0
0 0 0 0 0 4.37
C
Genetic Algorithms: an introduction
Comparison of results
0 modea
0o
90o
Genetic Algorithms: an introduction
Comparison of results
0 modes
0o
90o
Genetic Algorithms: an introduction
Comparison of results
0 modea
0 modes
Genetic Algorithms: an introduction
• Ordering/sequencing problems form a special type• Task is (or can be solved by) arranging some objects in a
certain order – Example: sort algorithm: important thing is which
elements occur before others (order)– Example: Travelling Salesman Problem (TSP) :
important thing is which elements occur next to each other (adjacency)
• These problems are generally expressed as a permutation:
– if there are n variables then the representation is as a list of n integers, each of which occurs exactly once
GA for Permutations
Genetic Algorithms: an introduction
The traveling salesman must visit every city in his territory exactly once and then return to the starting point; given the cost of travel between all cities, how should he plan his itinerary for minimum total cost of the entire tour?
TSP NP-Complete
The Traveling Salesman Problem (TSP)
Search space is BIG: for 30 cities there are 30! 1032 possible tours
Genetic Algorithms: an introduction
A vector v = (i1 i2… in) represents a tour (v is a permutation of {1,2,…,n})
Fitness f of a solution is the inverse cost of the corresponding tour
Initialization: use either some heuristics, or a random sample of permutations of {1,2,…,n}
We shall use the fitness proportionate selection
TSP (Representation, Initialization and Selection)
citiesn
Genetic Algorithms: an introduction
• Normal mutation operators lead to inadmissible solutions
– e.g. bit-wise mutation : let gene i have value j
– changing to some other value k would mean that k occurred twice and j no longer occurred
• Therefore must change at least two values
• Mutation parameter now reflects the probability that some operator is applied once to the whole string, rather than individually in each position
Mutation operations for permutations
Genetic Algorithms: an introduction
• Pick two allele values at random
• Move the second to follow the first, shifting the rest along to accommodate
• Note that this preserves most of the order and the adjacency information
Insert Mutation for permutations
Genetic Algorithms: an introduction
• Pick two alleles at random and swap their positions
• Preserves most of adjacency information (4 links broken), disrupts order more
Swap mutation for permutations
Genetic Algorithms: an introduction
• Pick two alleles at random and then invert the substring between them.
• Preserves most adjacency information (only breaks two links) but disruptive of order information
Inversion mutation for permutations
Genetic Algorithms: an introduction
• Pick a subset of genes at random
• Randomly rearrange the genes in those positions
(note subset does not have to be contiguous)
Scramble mutation for permutations
Genetic Algorithms: an introduction
Crossover builds offspring by choosing a sub-sequence of a tour from one parent and preserving the relative order of cities from the other parent and feasibility
Example:
p1 = (1 2 3 4 5 6 7 8 9) and
p2 = (4 5 2 1 8 7 6 9 3)
First, the segments between cut points are copied into offspring
o1 = (x x x 4 5 6 7 x x) and
o2 = (x x x 1 8 7 6 x x)
Crossover for TSP (ex.)
Genetic Algorithms: an introduction
Next, starting from the second cut point of one parent, the cities from the other parent are copied in the same order
The sequence of the cities in the second parent is 9 – 3 – 4 – 5 – 2 – 1 – 8 – 7 – 6
After removal of cities from the first offspring we get 9 – 3 – 2 – 1 – 8
This sequence is placed in the first offspring
o1 = (2 1 8 4 5 6 7 9 3), and similarly in the second
o2 = (3 4 5 1 8 7 6 9 2)
Crossover for TSP (ex.)
Genetic Algorithms: an introduction
Crossover for TSP (ex.)
Partially Mapped Crossover
Cycle crossover
Edge Recombination
…
Thank you