introduction to evolutionary algorithms yong wang lecturer, ph.d. school of information science and...
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Introduction to Evolutionary Algorithms
Yong Wang Lecturer, Ph.D.
School of Information Science and Engineering,Central South University
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Evolutionary Algorithms
Genetic Algorithm
Evolutionary Strategy
Evolutionary Programming
Particle Swarm Optimization
Differential Evolution
Outline of My Talk
The main branches of evolutionary algorithms
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Outline of My Talk
Evolutionary Algorithms
Genetic Algorithm
Evolutionary Strategy
Evolutionary Programming
Particle Swarm Optimization
Differential Evolution
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Why Evolutionary Algorithms (EAs)? (1/2)
-Search surface having multiple modals
The optimal solution Is it the optimal solution?
The optimal solution!
-Search surface having single modal
What is the difficult of the traditional optimization methods?
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Why Evolutionary Algorithms (EAs)? (2/2)
- How about searching from different directions? It’s a basic idea of EAs
The optimal solution!
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ant colony algorithmparticle swarm optimizationgenetic algorithm
What Are Evolutionary Algorithms?
• Evolutionary algorithms are intelligent optimization and search techniques inspired by nature
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The Framework of Evolutionary Algorithms
Population
Parent Set
Selection
the first individualthe second individual
the NPth individual
New Solutions
Crossover +Mutation
Replacement
xy
f(x,y)
020
4060
0
20
40
60-10
-5
0
5
10
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The Characteristics of Evolutionary Algorithms
• Search for the optimal solution from many points rather than one point
• Choose the individuals based on the fitness function and do not need the gradient information of the problems
• Use the random probability transition rule rather than the deterministic transition rule
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The Applications of Evolutionary Algorithms
• EAs can be used to solve different kinds of optimization problems
• For example:
unconstrained single-objective optimization problems
constrained single-objective optimization problems
unconstrained multi-objective optimization problems
constrained multi-objective optimization problems
EAs
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Unconstrained Single-objective Optimization Problems
min
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Constrained Single-objective Optimization Problems
13 14 15 16 170
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10
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x
y
feasible region
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Unconstrained Multi-objective Optimization Problems
min
-1 -0.5 0 0.5 1 1.5 2 2.5 30
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f(x)g(x)
min
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Constrained Multi-objective Optimization Problems
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The Top Journals in Evolutionary Computation Community
• IEEE Transactions on Evolutionary Computation (TEC, since 1997, 6 issues per year, about 60 papers per year)
• Evolutionary Computation (EC, since 1993, 4 issues per year, about 20 papers per year)
• IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics (SMCB)
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The Top Conferences in Evolutionary Computation Community
• IEEE Congress on Evolutionary Computation (CEC, Frequency: every year)
• Genetic and Evolutionary Computation Conference (GECCO, Frequency: every year)
• Evolutionary Multi-Criterion Optimization (EMO, Frequency: every two years)
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Outline of My Talk
Evolutionary Algorithms
Genetic Algorithm
Evolutionary Strategy
Evolutionary Programming
Particle Swarm Optimization
Differential Evolution
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Introduction (1/2)
• What are genetic algorithms?– Proposed by Professor J. Holland in the 1960s– Take their inspiration from Darwin’s theory of evolution, i.e.,
natural selection and survival of the fittest in the biological world (物竞天择,适者生存 )
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Introduction (2/2)
• The basic component of GAs– chromosome or string, which is also called an individual in a
population
0 0 1 1 12 0 1 0 1 1 1 0 1 0 0 241 1
string(chromosome)
character,Feature,(gene)
feature value(allele)
string position(locus)
Schema Highly fit, short-defining-length(BB)
Building Block
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Please note that the next six slides of GAs were provided by Dr. Chang Wook Ahn (http://www.evolution.re.kr/),
I just did some minor revisions.
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• The analogy between biological evolution and simple genetic algorithm
Simple Genetic Algorithm (1/4)
11100010000100001111
POPULATION
MATING POOL
NEW POPULATION
OFFSPRING
MATING (crossover)
MATES SELECTED
101010101011100010000011011100010000111111011000110011101010
1010101010111000100001000011110011101010
11100011110100001000
111000111101000010000001000011010000111111110000110000001000
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NONO
Simple Genetic Algorithm (2/4)
• Flowchart of a simple genetic algorithm
Define: Parameters Fitness function
Create population
Fitness evaluation
GA operators STOPYESConvergence Test
Selection + Crossover + Mutation
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Simple Genetic Algorithm (3/4)
• Three operators of the genetic algorithms1. Selection: Individuals are copied according to their fitness function values An artificial version of natural selection Roulette wheel selection, Tournament selection, etc.
101010101011100010000011011100010000111111011000110011101010
101010101000110111000100001111110110001100110111001101100011
Roulette wheel selection
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Simple Genetic Algorithm (4/4)
2. Crossover: Members of the newly reproduced strings in the mating pool are mated at random After choosing a cross site at random, the partial information of the two selected strings are exchanged3. Mutation: Changing a “1” to a”0” or visa versa, occurring points are randomly selected Escaping from the converging into local optimal solutions.
11100010000100001111
11100011110100001000
1110001000
1110101000
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Example of Genetic Algorithm
• Consider the following Complete Graph K4 (node 1 node 4)
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4433
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1234132413414
2
5
1
42
7
51167
1234123413414
1412341341234
1234123412341234
7565
1234123412341234
1234123412341234
5555
STOP
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4433
222
5
1
42
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Shortest Path
selection crossover selection
crossover
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Real-coded Genetic Algorithms
• Algorithmic framework Crossover operator
Steady-state genetic algorithms
B Q C
R
SelectionPlan
Generation Plan
ReplacementPlan
R’Update
Plan
① ②
③
④
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Crossover Operators (1/4)
• The offspring generated by unimodal normal distribution crossover (UNDX)
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Crossover Operators (2/4)
• The offspring generated by simplex crossover (SPX)
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Crossover Operators (3/4)
• The offspring generated by parent-centric recombination (PCX)
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Crossover Operators (4/4)
• The main characteristics of the crossover operators– UNDX and SPX are mean-centric recombination– PCX is parent-centric recombination– These three crossover operators are multi-parent crossover,
i.e., unlike the common crossover operators, all of them need more than two parents to take part in crossover.
• These three crossover operators are the most competitive crossover operators in real-coded genetic algorithms.
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Outline of My Talk
Evolutionary Algorithms
Genetic Algorithm
Evolutionary Strategy
Evolutionary Programming
Particle Swarm Optimization
Differential Evolution
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Evolutionary Strategy (ES)
• ES was proposed by I. Rechenberg in 1964• There are several versions of ES, for instance
– ( )-ES
– ( )-ES
,
offspringµ parents µ parents
µ parents and offspring
µ parents µ parents
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The Framework of ( )-ES
2 2,x
,x
1 1,x
2 2,x
,x
1 1,x
2 2,x
,x
1 1,x
Algorithmic framework
k=1,…,
objective function
value
,
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Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES)
• In addition to ( )-ES and ( )-ES, two advanced versions of ES have been proposed by N. Hansen– CMA-ES– Restart CMA-ES (R-CMA-ES)
• Currently, CMA-ES is the most well-known ES and R-CMA-ES is the most competitive ES
http://www.lri.fr/~hansen/index.html
,
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The Main Idea of CMA-ES
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Outline of My Talk
Evolutionary Algorithms
Genetic Algorithms
Evolutionary Strategy
Evolutionary Programming
Particle Swarm Optimization
Differential Evolution
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Introduction to the EP
• EP was proposed by L. J. Fogel in 1966
2 2,x
,x
1 1,x
2 2,x
,x
1 1,x
objective function
value
2 2,x
,x
1 1,x
Algorithmic framework
objective function
value
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Cauchy Mutation VS Gaussian Mutation
)1,0()()()(' jjii Njjxjx
jjii jjxjx )()()('
X. Yao, Y. Liu and G. Lin, ``Evolutionary programming made faster,'' IEEE Transactions on Evolutionary Computation, vol. 3, no. 2, pp. 82-102, 1999.
Gaussian mutation
Cauchy mutation
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Cauchy Distribution VS Gaussian Distribution
amplitude
tail
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Theoretical Background (1/3)
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Theoretical Background (2/3)
Mean value theorem of integrals
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Theoretical Background (3/3)
absolute value
Remark: the similar analysis can be carried out for Cauchy distribution
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Outline of My Talk
Evolutionary Algorithms
Genetic Algorithm
Evolutionary Strategy
Evolutionary Programming
Particle Swarm Optimization
Differential Evolution
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• Particle Swarm Optimization (PSO) was invented by James Kennedy and Russ Eberhart in 1995
• PSO takes inspiration of the motion of a flock of birds• PSO has been used to solve many kinds of problems• In PSO, each potential solution is regarded as a particl
e.
Particle Swarm Optimization
J. Kennedy R. Eberhart
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The Movement Equations of the Particles
• w denotes the inertia weight, c1 and c2 are the acceleration constants, r1 and r2 are two separately generated uniformly distributed random numbers in the range [0,1].
• denotes the jth variable of the ith particle at generation t.
xi,jt
pbesti,jt
gbesti,jt
vi,jt
vi,jt+1
xi,jt+1
the personal best the best of the swarm
the movement of each variable
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Analysis of PSO
1 * ( )t t tv v x 1 1t t tx x v
1 1 1* ( )t t tx v x
11 1 2 2* * *( ) * * ( )t t t t t tv v c r pbest x c r gbest x
1 1t t tx x v
1 1 1c r 2 2 2c r
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Let
2 1 2t t tx x v
1 1 1*( ) ( )t t t tx x x x 1(1 ) t tx x
W. Hu, Z. LI. A simpler and more effective particle swarmoptimization algorithm, Journal of Software, 2007,18(4): 861-868.
1 2
1 2
t tpbest gbest
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Swarm Topology
• Since the particles interact with each other, the swarm topology is very important for the performance of PSO
R. Mendes, J. Kennedy, and J. Neves. The fully informed particleswarm: simpler, maybe better. IEEE Transactions on Evolutionary Computation, vol. 8, no. 3, pp. 204-210, 2004.
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Outline of My Talk
Evolutionary Algorithms
Genetic Algorithm
Evolutionary Strategy
Evolutionary Programming
Particle Swarm Optimization
Differential Evolution
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Differential Evolution (1/2)
• Differential evolution (DE), proposed by Storn and Price in 1995, is one of the main branches of evolutionary algorithms (EAs).
• In principle, DE is a kind of real-coded greedy genetic algorithm (GA).
• DE includes three main operators, i.e., mutation operator, crossover operator, and selection operator.
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Differential Evolution (2/2)
• The algorithmic framework of DE
Remark: mutation + crossover = trial vector generation strategy
the target vector
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The Mutation Operators
• rand/1 )( ,3,2,1, GrGrGrGi xxFxv
• rand/2 )()( ,5,4,3,2,1, GrGrGrGrGrGi xxFxxFxv
)( ,2,1,, GrGrGbestGi xxFxv
• best/1
)()( ,2,1,,,, GrGrGiGbestGiGi xxFxxFxv
)()( ,4,3,2,1,, GrGrGrGrGbestGi xxFxxFxv
• best/2
• current-to-best/1
• current-to-rand/1 )()( ,3,2,,1,, GrGrGiGrGiGi xxFxxrandxv
Remark: r1, r2, r3, r4, and r5 are different indexes uniformly randomly selected from , is the best individuals in the current population, and F is the scaling factor.
}{\},,1{ iNP
the base vector the difference vector
Gbestx ,
the fashion the base vector has been selected
the number of the difference vector
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The Characteristics of the Mutation Operators (1/3)
• rand/1 )( ,3,2,1, GrGrGrGi xxFxv
• Characteristics– “rand/1” is the most commonly used mutation operator in the
literature. – All vectors for mutation are selected from the population at
random and, consequently, it has no bias to any special search directions and chooses new search directions in a random manner.
– It usually demonstrates slow convergence speed and bears stronger exploration capability.
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The Characteristics of the Mutation Operators (2/3)
• rand/2 )()( ,5,4,3,2,1, GrGrGrGrGrGi xxFxxFxv
• Characteristics– In “rand/2”, two difference vectors are added to the base
vector, which might lead to better perturbation than the strategies with only one difference vector.
– It can generate more different trial vectors than the “rand/1” mutation operator.
– When using “rand/2”, the diversity of the population can be kept, however, it has a side effect on the convergence speed of DE.
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The Characteristics of the Mutation Operators (3/3)
)( ,2,1,, GrGrGbestGi xxFxv
• best/1
)()( ,2,1,,,, GrGrGiGbestGiGi xxFxxFxv
)()( ,4,3,2,1,, GrGrGrGrGbestGi xxFxxFxv
• best/2
• current-to-best/1
• Characteristics – “best/1”, “best/2” and “current-to-best/1”, usually have the fast c
onvergence speed and perform well when solving unimodal problems.
– They are easier to get stuck at a local optimum and thereby lead to a premature convergence when solving multimodal problems.
– The “best/1” is a degenerated case of the“current-to-best/1” with the first scaling factor F being equal to 1.
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The Crossover Operators (1/2)
• Binomial crossover
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The Crossover Operators (2/2)
• Exponential crossover
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The Characteristics of the Crossover Operators
• Characteristics– Binomial crossover is similar to discrete crossover in GA. – Exponential crossover is functionally equivalent to two-point
crossover in GA.– Exponential crossover has the capability in maintaining the
linkage among variables and the building block. – Binomial crossover may destroy building block.
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DE Variations
• By combining different mutation operators and different crossover operators, we can obtain different DE variants.
• DE/x/y/z– DE: differential evolution– x: the fashion the base vector has been selected– y: the number of the difference vector– z: the type of the crossover operator; “bin” represents the bi
nomial crossover and “exp” represents the exponential crossover
• DE/rand/1/bin, DE/rand/1/exp, DE/rand/2/bin, …
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The illustrative graph of DE/rand/1/bin
the triangle denotes the trial vector Giu ,
Gix ,
Grx ,2
Grx ,1
Giv ,
0
Grx ,3
1x
2x
)( ,3,2 GrGr xxF
base vector
perturbed vectors
Welcome to visit my homepage: http://deptauto.csu.edu.cn/staffmember/YongWang.htm