introduction to evolutionary algorithms yong wang lecturer, ph.d. school of information science and...

Introduction to Evolutionary Algorithms

Yong Wang Lecturer, Ph.D.

School of Information Science and Engineering,Central South University

ywang@csu.edu.cn

2

Evolutionary Algorithms

Genetic Algorithm

Evolutionary Strategy

Evolutionary Programming

Particle Swarm Optimization

Differential Evolution

Outline of My Talk

The main branches of evolutionary algorithms

3

Outline of My Talk

Evolutionary Algorithms

Genetic Algorithm

Evolutionary Strategy

Evolutionary Programming

Particle Swarm Optimization

Differential Evolution

4

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?

5

Why Evolutionary Algorithms (EAs)? (2/2)

- How about searching from different directions? It’s a basic idea of EAs

The optimal solution!

6

ant colony algorithmparticle swarm optimizationgenetic algorithm

What Are Evolutionary Algorithms?

• Evolutionary algorithms are intelligent optimization and search techniques inspired by nature

7

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

8

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

9

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

10

Unconstrained Single-objective Optimization Problems

min

11

Constrained Single-objective Optimization Problems

13 14 15 16 170

5

10

15

20

x

y

feasible region

12

Unconstrained Multi-objective Optimization Problems

min

-1 -0.5 0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

7

8

9

f(x)g(x)

min

13

Constrained Multi-objective Optimization Problems

14

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)

15

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)

16

Outline of My Talk

Evolutionary Algorithms

Genetic Algorithm

Evolutionary Strategy

Evolutionary Programming

Particle Swarm Optimization

Differential Evolution

17

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 )

18

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.

19

• 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

20

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

21

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

22

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

23

Example of Genetic Algorithm

• Consider the following Complete Graph K4 (node 1 node 4)

11

4433

22

1234132413414

2

5

1

42

7

51167

1234123413414

1412341341234

1234123412341234

7565

1234123412341234

1234123412341234

5555

STOP

11

4433

222

5

1

42

7

Shortest Path

selection crossover selection

crossover

24

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

25

Crossover Operators (1/4)

• The offspring generated by unimodal normal distribution crossover (UNDX)

26

Crossover Operators (2/4)

• The offspring generated by simplex crossover (SPX)

27

Crossover Operators (3/4)

• The offspring generated by parent-centric recombination (PCX)

28

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.

29

Outline of My Talk

Evolutionary Algorithms

Genetic Algorithm

Evolutionary Strategy

Evolutionary Programming

Particle Swarm Optimization

Differential Evolution

30

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

31

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

,

32

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

,

33

The Main Idea of CMA-ES

34

Outline of My Talk

Evolutionary Algorithms

Genetic Algorithms

Evolutionary Strategy

Evolutionary Programming

Particle Swarm Optimization

Differential Evolution

35

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

36

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

37

Cauchy Distribution VS Gaussian Distribution

amplitude

tail

38

Theoretical Background (1/3)

39

Theoretical Background (2/3)

Mean value theorem of integrals

40

Theoretical Background (3/3)

absolute value

Remark: the similar analysis can be carried out for Cauchy distribution

41

Outline of My Talk

Evolutionary Algorithms

Genetic Algorithm

Evolutionary Strategy

Evolutionary Programming

Particle Swarm Optimization

Differential Evolution

42

• 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

43

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

44

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

21

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

45

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.

46

Outline of My Talk

Evolutionary Algorithms

Genetic Algorithm

Evolutionary Strategy

Evolutionary Programming

Particle Swarm Optimization

Differential Evolution

47

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.

48

Differential Evolution (2/2)

• The algorithmic framework of DE

Remark: mutation + crossover = trial vector generation strategy

the target vector

49

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

50

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.

51

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.

52

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.

53

The Crossover Operators (1/2)

• Binomial crossover

54

The Crossover Operators (2/2)

• Exponential crossover

55

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.

56

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, …

57

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