Transcript Genetic algorithms

Genetic algorithms: an
introduction
Artem Eremin,
j. researcher, IMMI KSU
Kuban State University
Krasnodar, Russia
Motivation
Genetic Algorithms: an introduction
Motivation
Genetic Algorithms: an introduction
Motivation
experimental data
Material properties (Cij) - ???
• Doppler laservibrometry for measuring
out-of-plane velocities
Wavelet transform
TOF
• Time-of-Flight (TOF) with wavelet
transform

g
d
c 
TOF
min F  ?
Cij
Np
F :  (cg , j  cg , j )2
j 1
Genetic Algorithms: an introduction
Optimization
“Optimization is the process of making something better”
Every day we subconsciously
solve some optimization problems!
Genetic Algorithms: an introduction
Optimization
Genetic Algorithms: an introduction
Minimum-seeking algorithms
1. Exhaustive Search = Brute Force
2. Analytical Optimization
2 f
f (x)  0,
?
x
3. Nelder-Mead downhill Simplex Method
4. Optimization based on Line Minimization (the
coordinate search method, the steepest
descent algorithm, Newton’s method, DavidonFletcher-Powell (DFP) algorithm, etc …)
Genetic Algorithms: an introduction
Minimum-seeking algorithms
1 – 4 can converge to a local minimum!
Natural optimization methods
Not the panacea, but …
Simulated annealing
(Kirkpatrick et al., 1983)
Particle swarm optimization
(Parsopoulos and Vrahatis, 2002)
Genetic algorithms
(Holland, 1975)
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…
f ( x, y)  y sin 4 x  1.1x sin 2 y
0  x, y  10
Phenotype space
Genotype space =
{0,1}L
Encoding
(representation)
10
y
10010001
10010010
010001001
011101001
0
x 10
Decoding
(inverse representation)
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
x
Chromosome – 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
chromosome  [ p1, p2 , ... , pNvar ]
cost  f (chromosome)
N pop chromosomes
Genetic Algorithms: an introduction
How does it works?
So…
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
Genetic Algorithms: an introduction
How does it works?
The Evolutionary Cycle
selection
parents
modification
modified
offspring
initiate &
evaluate
population
evaluation
evaluated offspring
deleted
members
Or so…
discard
Genetic Algorithms: an introduction
Generation of the initial population
pi [ai , bi ]
pi  ai  random[0,1]  (bi  ai )
i  1, Nvar
Coding: 4.25  01101101...
Ex.
Npop=6
s1 = 1111010101
s2 = 0111000101
s3 = 1110110101
s4 = 0100010011
s5 = 1110111101
s6 = 0100110000
Genetic Algorithms: an introduction
f (s1) = 7
f (s2) = 5
f (s3) = 7
f (s4) = 4
f (s5) = 8
f (s6) = 3
Selection
N keep  X rate N pop
We are kind!
Let’s save everybody!
X rate  [0.4, 0.6]
or
i, fi  ftr  save!
Mating pool
Genetic Algorithms: an introduction
Selection
Nkeep , N pop  Nkeep
1. Pairing from top to bottom
2. Random pairing
1  N keep
2  N keep  1
...
sc ,1  random[1, Nkeep ]
sc ,2  random[1, Nkeep ]
3. Weighted random pairing
roulette wheel weighting
Genetic Algorithms: an introduction
Selection
The roulette wheel method:
Individual i will have a probability
f (i )
 f (i) to be chosen
i
1
n
Area is
proportional to
fitness value
2
We repeat the extraction as many
times as it is necessary
3
4
Genetic Algorithms: an introduction
Selection
4. Tournament selection
1. Zenit
2. CSKA
…
16. Krylia Sovietov
50
48
14
a) randomly pick a small subset
b) perform a “tournament”
c) “the winner takes it all”
Tournament + Threshold = No SORTING!!!
Genetic Algorithms: an introduction
Mating (Crossover)
Simple 1-point crossover
• Choose a random
point on
the two parents
• Performance
with 1 Point
Crossover
depends
• Split parents at this crossover point
on the order that variables occur in the
• Create children by exchanging tails
representation
• P typically in range (0.6, 0.9)
c
– 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 / mainbackground
• 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
Real valued problems
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
bi  ai L1
(a1 ,..., aL )  ai  L
 ( aL j  2 j ) [ai , bi ]
2  1 j 0
• Only 2L values out of infinite are represented
• L determines possible maximum precision of solution
• High precision  long chromosomes (slow evolution)
Genetic Algorithms: an introduction
Floating point mutations
General scheme of floating point mutations
s  ( p1, p2 ,..., pN pop )  s '  ( p '1, p '2 ,..., p 'N pop )
pi , p 'i ai , bi 
• Uniform mutation:
p 'i drawn randomly (uniform) from ai , bi 
• Analogous to bit-flipping (binary) or random resetting
(integers)
Genetic Algorithms: an introduction
Floating point mutations
• 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 + )
Genetic Algorithms: an introduction
Crossover for real valued GAs
• 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
Genetic Algorithms: an introduction
Single arithmetic crossover
•
•
•
Parents: x1,…,xn  and y1,…,yn
Pick a single gene (k) at random,
child1 is:
•
reverse for other child. e.g. with  = 0.5
x1, ..., xk ,   yk  (1  )  xk , ..., xn
Genetic Algorithms: an introduction
Simple arithmetic crossover
•
•
•
Parents: x1,…,xn  and y1,…,yn
Pick random gene (k) after this point mix values
child1 is:
x , ..., x ,   y
 (1   )  x
, ...,  y  (1   )  x
1
k
k 1
k 1
n
n
•
reverse for other child. e.g. with  = 0.5
Genetic Algorithms: an introduction
“Whole” arithmetic crossover
•
•
•
Most commonly used
Parents: x1,…,xn  and y1,…,yn
child1 is:
a  x  (1  a)  y
•
reverse for other child. e.g. with  = 0.5
Genetic Algorithms: an introduction
micro-GA
First generation (random values)
Tournament selection
SBX crossover
Select fittest
individual
Start new
generation
Good results?
Enough iterations?
No
Genetic Algorithms: an introduction
Yes
result
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
Genetic Algorithms: an introduction
Examples
Experimental specimens
4 CFRP–plates
H  2, 25mm
[0o ,0o ,0o ,0o ]
H  2,35mm
[0o ,90o ,90o ,0o ]
Vf  60%
Vf  58%
Genetic Algorithms: an introduction
Material properties
Ex  107  2% GPa;
E y  Ez  8.9  2% GPa
Gyz  2.82  2% GPa; Gxz  Gyz  4.38  1% GPa
 xz   xy  0.25  0.32;  yz  0.49  0.56
110.5

 7.0
 7.0
C 
0
0

0
7.0
7.0
0
0
13.8
8.2
0
0
8.2
13.8
0
0
0
0
2.8
0
0
0
0
4.37
0
0
0
0
Genetic Algorithms: an introduction


0 
0 
 GPa
0 
0 

4.37 
0
Comparison of results
o
0
a0 mode
90 o
Genetic Algorithms: an introduction
Comparison of results
o
0
s0 mode
90 o
Genetic Algorithms: an introduction
Comparison of results
a0 mode
s0 mode
Genetic Algorithms: an introduction
GA for Permutations
•
•
•
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
Genetic Algorithms: an introduction
The Traveling Salesman Problem (TSP)
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
Search space is BIG:
for 30 cities there are
30!  1032 possible tours
Genetic Algorithms: an introduction
TSP (Representation, Initialization and Selection)
n cities
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
Genetic Algorithms: an introduction
Mutation operations for permutations
• 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
Genetic Algorithms: an introduction
Insert Mutation for permutations
• 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
Genetic Algorithms: an introduction
Swap mutation for permutations
• Pick two alleles at random and swap their
positions
• Preserves most of adjacency information
(4 links broken), disrupts order more
Genetic Algorithms: an introduction
Inversion mutation for permutations
• 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
Genetic Algorithms: an introduction
Scramble mutation for permutations
• Pick a subset of genes at random
• Randomly rearrange the genes in those
positions
(note subset does not have to be contiguous)
Genetic Algorithms: an introduction
Crossover for TSP (ex.)
Crossover builds offspring by choosing a subsequence 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)
Genetic Algorithms: an introduction
Crossover for TSP (ex.)
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)
Genetic Algorithms: an introduction
Crossover for TSP (ex.)
Partially Mapped Crossover
Cycle crossover
Edge Recombination
…
Genetic Algorithms: an introduction
Thank you