Evolutionary Optimization Wi th focus on genetic algorithm and regarding applications

Evolutionary Optimization

With focus on genetic algorithm and regarding applications

Danial Khashabi ([email protected])Amirkabir University of Technology, School of Electrical EngineeringJuly 22, 2010

Introduction to

In memory of Caro Lucas, Valuable Iranian Scientist

Lecture Overview:• Evolutionary optimization.

– What is EO in general.• Genetic Algorithms.

– Overview of search methods.– Some examples of evolutionary optimization for getting

perception!– Brief history of Digital Genetics.– Introducing genetic algorithm members.

• Steps for solving a problem using genetic algorithm, a detailed view.• A review on genetic algorithm.• Introducing applications of GA.• Some examples:

– Optimizing a one-variable function.– Optimizing a two-variable function.– Traveling Salesman problem(TSP).– Evolving hardware using GA.

Evolutionary Optimization• Solution for problem is formed by “Population” • Population consists of individuals.• Individuals that are more fitter, have more chance to survive! • Solutions are evolved in every generation.• Every population is parent generation for next generation.• Fitness in population grows gradually, as generations pass.

– This is called “Evolution”!

[“Evolutionary Algorithms”: S.N.Razavi]

What is EO in general?• It’s a branch of Computation Theory in Computer

Science?– So why an engineer needs to know about EO?– It is an optimization method and it can be applied

to bunch of problems!• It is inspired from Darwin's “Evolution Theory”.• Genetic algorithms are a part of evolutionary

computing, which is a rapidly growing area of artificial intelligence.

[Charles Darwin: 1809-1882 : http://en.wikipedia.org/wiki/Charles_Darwin]

Genetic Algorithm• A way to employ evolution in solutions• Why evolution?!

– Man computational problems require searching through a large possibilities for solutions.

• Search and Optimization– Based of variation and selection– by understanding the adaptive processes of natural systems

• Search for ?! – Find a better solution to a problem in a large space.

• What is a better solution? – A good solution is specified by “Fitness Function”!

[http://media.brainz.org/uploads/2009/02/genetic-algorithms.jpg]

Overview of Search Methods

F inon acc i N ew ton

D irec t m ethods Indirec t m ethods

C alcu lus-based tech n iques

Evolu tionary s trategies

C entra l ized D is tributed

Para l le l

S tea dy-s ta te G enera tiona l

Sequentia l

G ene tic a lgor ithm s

Evolutionary a lgorithm s Sim u lated an nealing

G uide d random sea rch techniques

D ynam ic program m in g

Enu m erative techn iques

Sea rch tec hniques

You are here !

[Genetic Algorithms: A Tutorial: W.Wliliams]

Local vs. Global; a BIG challenge!

• This an important challenge !

[Optimization with Genetic Algorithm/Direct Search Toolbox : Ed Hall]

GA vs. Some other search methods

• Search methods: – Aggressive methods(e.g. Simulated Annealing)

• Can be trapped in local minima• Initial position is important

– Non-Aggressive methods(e.g. GA)• Traces Global minima• Can not guarantee discovery of hilltop

[Genetic Algorithms: A.Dix, M.Sifalakis]

One example !

[Optimization with Genetic Algorithm/Direct Search Toolbox, E.Hall]

One example !

[Optimization with Genetic Algorithm/Direct Search Toolbox, E.Hall]

Brief History

• Evolutionary computing – was introduced in the 1960s – by I. Rechenberg – in his work "Evolution strategies"

• Genetic Algorithms (GAs) – invented by John Holland – This lead to Holland's book "Adaption in Natural

and Artificial Systems" published in 1975. • Genetic Programming?!• In 1992 John Koza has used genetic algorithm to

evolve programs to perform certain tasks. He called his method "genetic programming" (GP).

Genetic Algorithm• A chromosome is a string representation of a candidate solution to

a problem. – Bin(0/1) 101000111010111010111– Decimal(0-9) 345304539475330394537– Alfa(A-Z) ABBCBDCCCBBDBAAABCA– Hex(0-F): 982234BA68AB23634FDD

• The genes are single or subset of digits at specific locations in the chromosome string:

Bin 0101000100100111101 • An Allele is possible values a gene can take; e.g. 1/0 for binary.

Gene

Solving a Problem using GAStep 1: Encoding

• We need to encode the problem in a proper manner.• Solutions are coded as “chromosomes”

– Algorithm only understands numbers.– Improper coding may result inefficient solutions.– Some times a new method of coding may result in

new methods and solutions. – It’s a kind of Heuristic.

• Encoding depends on the problem heavily.• This could be:

– String of real numbers.– String of 1 and 0(binary coding)

• This is the most common coding.• This is a simple encoding and also powerful and fast.

Chromosome A 101100101100101011100101

Chromosome B 111111100000110000011111


• Example1:

• Example2:

– Solution for 23, ' 6+5*4/2+1' would be represented like so:

0110 1010 0101 1100 0100 1101 0010 1010 00016 + 5 * 4 / 2 + 1

– These genes are all strung together to form the chromosome: 011010100101110001001101001010100001

- The possible genes 1110 & 1111 will remain unused and will be ignored by the algorithm if encountered.

1 0 1 1 0 1 0 0 0 1 0 1 1

Size Shape Speed

A Sample Chromosome

[Genetic Algorithms: Dr.K.Kiani]

Decimal

Binary Coded Decimal(BCD)

0 00001 00012 00103 00114 01005 01016 01107 01118 10009 1001+ 1010- 1011* 1100/ 1101


• Example:

• Every encoding needs a decoding(After Optimization)– Expected coding is :

number -> operator -> number -> operator– We may encounter some conditions that is not predicted based on encoding

algorithm:

• The best encoding is which:– Probability of encountering an undefined condition is low– By means of least bits it represents the most information– It covers whole search space!

• We must define a rule how to deal with unexpected conditions.– By the above rule we can interpret above chromosome as:

2 + 7[Genetic Algorithms: Dr.K.Kiani]

0010 1010 0011 1100 0101 1010 0110 2 + 3 * 5

+ 6=30

1000 1100 0011 1101 0011 1010 0010 8 * 3 / 3

+ 2=10

0010 0010 1010 1110 1011 0111 0010 2 2 + n/a -

7 2


• Direct value encoding for real numbers: – In the value encoding, every chromosome is a sequence of some values

Example of Problem: Finding weights for a neural networkThe problem: A neural network is given with defined architecture. Find

weights between neurons in the neural network to get the desired output from the network.

Encoding: Real values in chromosomes represent weights in the neural network.


Chromosome A 1.2324 5.3243 0.4556 2.3293 2.4545

Chromosome C (back), (back), (right), (forward), (left)

Solving a Problem using GAStep 2: GA Operators

• Crossover(Recombination): – Operate on selected parent chromosomes to create new

offspring– decomposes two distinct solutions (chromosomes) and then

randomly mixes their parts to form novel solutions (chromosomes)

• Mutation:– A surprisingly small role is played by mutation.– Randomly changes the offspring.– This creates diversity in search space. – Prevent falling of all solutions into a local optimum.– Most mutations are damaging rather than beneficial.

• therefore, rates must be low to avoid the destruction of species.


• Crossover:– Example(Single Point Crossover)

– Example(Two- Point Crossover)

– Example(Simple Arithmetic Crossover)• Parents :

• Pick random gene (k) after this point mix values


Parent1

Parent2 1011011111

1010000000 1011011111

1010000000Offspring2

Offspring1Random

1011011111

1010000000Parent1

Parent2

Random

1010011111

1011000000Offspring1

Offspring2

nxx ..., ,1

nxkxkykxx )1(ny ..., ,1)1(1 , ..., ,1 nyy ..., ,1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.3

0.1 0.4 0.5 0.3 0.2 0.6 0.6 0.3Parent 1:Parent 2:

Offspring 1:

Offspring 2:

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.46.064.016.048.08.0)8.01(6.08.0

0.3 0.1 0.4 0.5 0.3 0.2 0.6 0.8 0.8

4.042.018.024.09.0)8.01(3.08.0

8.076.012.064.06.0)8.01(8.08.0 8.078.006.072.03.0)8.01(9.08.0


• Mutation:– Example:

• Example( Order Changing Mutation):• two numbers are selected and exchanged

• Example(Adding Mutation)• a small number (for real value encoding) is added to

(subtracted from) selected values


1010000000Parent Offspring

Random

1010000100

mutated

(1 2 3 4 5 6 8 9 7) => (1 8 3 4 5 6 2 9 7)

(1.29 5.68 2.86 4.11 5.55) => (1.29 5.68 2.73 4.22 5.55)

Solving a Problem using GAStep 3: Start Optimization!

• Lets have a look on optimization process:

A random population of chromosomes are generated.(Initial Population)

Repeat {The fitness function is applied to each chromosome.(Evaluation)Selection is applied in a favorable manner.(Selection)Crossover is applied on pairs to generate new population. Mutation is applied to some offspring. }

• Generating Initial Generation:– Most of time it’s random– Initial population must be diverse

Solving a Problem using GAStep 3: Selection(Reproduction)

• New generation is produced based on generation’s fitness value.• Must make sure that fittest chromosomes are propagated to the next generation.• One popular selection: Roulette-wheel selection method:

– Every individual has a probability(chance) to be selected proportional proportional to its fitness value.

– So better individuals(more fitted individuals) have more chance to be selected! – Even less fitted solutions have chance to be selected !

• This is a good characteristic ! • Causes more diversity on population. • As a result algorithm avoids of trapping in local minimum.

[http://www.edc.ncl.ac.uk/assets/hilite_graphics/rhjan07g02.png]

Solving a Problem using GAA Review on Algorithm:

[Genetic Algorithms: A.Dix, M.Sifalakis]

A Review on Characteristics:

• Stochastic• Used for -> hard problems• Maintains -> population• Every Solution -> chromosome • Reproduction -> new population • Better solutions -> more chance of

survival

Let see an examples!: one variable function(Slide1)

• Optimizing a one variable function:– If– The goal is to

• Because function is determined explicitly its possible to determine maximum of function by gradient:

• Numerical result:

0.1).10sin(.)( xxxf

-1 -0.5 0 0.5 1 1.5 2-1

-0.5

0

0.5

1

1.5

2

2.5

3


)(maxarg]2,1[

xfx

xxxxxxf

.10).10tan(0).10cos(.10).10sin()(

85.20.1)2

18sin(.85.1)85.1(

85.12037

19

f

x

)(max]2,1[xf

x


• Lets again review: What we exactly need to find maximum of a one variable function:- Find a proper encoding for solutions- A proper GA operator algorithms(Mutation and Crossover).- A proper selection method.Now we are going to discuss each one: • Encoding:

• Goal to encode real numbers [-1,2] using binary string: 111010….01• I assume this corresponding:

• The length of chromosome depends on required precision.• If I need N digit of accuracy in real numbers space: (Say N is 6)

• If I don’t care decimal point, I have:

• So by having a 22-digit length string it is possible to compute a rang of real numbers with 6-precision, specifically [-1,2]

• For finding corresponding real number from binary string, if chromosome is•

000 ... 0

111 ... 1-12

41943042300000022097152 2221

202021 )( bbb

312

2.0.1 22

10

21

0

i

iib

x


• Example of a chromosome:

• For simplicity, we use simplest(one-point) crossover and mutation methods. • Fitness Function:

– It is clear that a clear that a chromosome with higher f(x) is better! – So we can assume that: where x is real part of

chromosome.– E.g:


Chromosome: 10001011101101010001112288967)1111010100011000101110( 2

637197.04194303

3.22889670.1 x

)()( xfvnctionFittnessFu

627888.1)0111111100011110000000(958973.0)0000000001000000001110(

637197.0)1111010100011000101110(

33

22

11

xtodcorresponevxtodcorresponevxtodcorresponev

250650.2)()(078878.0)()(

586345.1)()(

33

22

11

xfvFitnessxfvFitnessxfvFitness


• For simplicity, we use simplest(one-point) crossover and mutation methods.

• Mutation:

• Crossover:


721638.1)0111111100011110100000(

627888.1)0111111100011110000000(

3'

3'

33

xtodcorresponev

xtodcorresponev

082257.0)()(

250650.2)()(

3'

3'

33

xfvfitness

xfvfitness

627888.1)0111111100011110000000(958973.0)0000000001000000001110(

33

22

xtodcorresponevxtodcorresponev

250650.2)()(078878.0)()(

33

22

xfvfitnessxfvfitness

666028.1)0000000001001110001110(

998113.0)0111111100010000000000(

3'

3'

2'

2'

xtodcorresponev

xtodcorresponev459245.2)()(

940865.0)()(

3'

3'

2'

2'

xfvfitness

xfvfitness


• Results: – If populationSize = 50– P_crossover = 0.25– P_mutation = 0.01

• Note about improvement of best answer.• After 150 generations:

• Remember result from explicit method:


Generation number

Fitness function

1 1.4419426 2.2500038 2.2502839 2.250284

10 2.25036312 2.32807739 2.34425140 2.34508751 2.73893099 2.849246137 2.850217145 2.850227

850773.1)0100010000011111001101(

max

max

xtodcorresponev

85.20.1)2

18sin(.85.1)85.1(

85.12037

19

f

x

Another examples!A 2-variable function(Slide1)

• If

• Again we need to decode x1 and x2 like previous example.• As mentioned number of digits in chromosomes depends on desired

precision.


).20sin(.).4sin(.5.21),( 221121 xxxxxxf 8.51.41.123 21 xandx


• Domain of x1 is 12.1-(-3)=15.1– 4 digits of precision: – So we need 18-length bit string for representing x1

• Doman of x2 is 5.8-4.1=1.7– 4 digits of precision: – So we need 15-length bit string for representing x2

• We can assume that out chromosome has length 15+18=33• Example chromosome: 010001001011010000111110010100010• If we want to decode? E.g.: above chromosome.

So above chromosome corresponds to (x1, x2)= (1.052426,5.755330)And the fitness value for this point is: f(1.052426,5.755330) =

20.252640 [Genetic Algorithms: Dr.K.Kiani]

).20sin(.).4sin(.5.21),( 221121 xxxxxxf 8.51.41.123 21 xandx

26214421510002131072 1817

32768217000216384 1514

052426.1262143

1.15.703520.312

)0.3(1.12).110100000100010010(0.3 1821

decimalx

755330.532767

7.1.319061.4121.48.5).00101111001010(1.4 1522

decimalx


• Initial Population:– populationSize = 20– Generated randomly:


110100010111110011110011011100101100000011000111111111110001010100101011111010100001110100111010000101000110111111001111010110101111011000110100100011111101100111100110001111101101110000101110111011001011110010100001010100100100110110000000100010001000111110111110000010101000011101101001101000101000000110111110110101100110011111011101000011100110000010000011110100000000111101100010110100000001111011010110001110100011000011000011101101111100110101110010001000011101011111110010101000001010000101011111100001010011010001110110010100000111001101001111000110001101101011101111001000000000100000010101010001100110111001110001001111101001101100001111111001101000

20

19

18

17

16

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

vvvvvvvvvvvvvvvvvvvv


• Evaluating by fitness fun:


7768.3876669.13),(),(|7573.49359.7|1101000101111100111100110111001010959.20),(),(|3955.57466.1|1000000110001111111111100010101004141.15),(),(|1582.58430.3|1010111110101000011101001110100006961.13),(),(|5713.43675.3|1010001101111110011110101101011118672.23),(),(|9937.42115.9|0110001101001000111111011001111000602.30),(),(|0545.50889.11|1100011111011011100001011101110118762.19),(),(|1513.53359.1|0010111100101000010101001001001103167.27),(),(|3786.51346.11|1100000001000100010001111101111100116.15),(),(|2394.43561.9|0000101010000111011010011010001014106.23),(),(|9960.41300.3|0000001101111101101011001100111112784.21),(),(|7937.45548.2|0111010000111001100000100000111101277.16),(),(|3814.57954.0|1000000001111011000101101000000019597.17),(),(|7030.49106.4|1110110101100011101000110000110000208.16),(),(|5802.59914.0|0111011011111001101011100100010001004.18),(),(|3919.48117.1|0111010111111100101010000010100003411.25),(),(|7344.42551.1|1010111111000010100110100011101104067.17),(),(|5934.52786.5|010100000111001101001111000110001

5263.19),(),(|3903.45166.2|1011010111011110010000000001000005800.7),(),(|3802.43484.10|0101010100011001101110011100010010196.26),(),(|6522.50844.6|111101001101100001111111001101000

21212120

21212119

21212118

21212117

21212116

21212115

21212114

21212113

21212112

21212111

21212110

2121219

2121218

2121217

2121216

2121215

2121214

2121213

2121212

2121211

FitnessTotalxxfxxfitnessxxvxxfxxfitnessxxvxxfxxfitnessxxvxxfxxfitnessxxvxxfxxfitnessxxvxxfxxfitnessxxvxxfxxfitnessxxvxxfxxfitnessxxvxxfxxfitnessxxvxxfxxfitnessxxvxxfxxfitnessxxvxxfxxfitnessxxvxxfxxfitnessxxvxxfxxfitnessxxvxxfxxfitnessxxvxxfxxfitnessxxvxxfxxfitnnessxxvxxfxxfitnessxxvxxfxxfitnessxxvxxfxxfitnessxxv

It is clear, that the chromosome v15 is the strongest one, and the chromosome v2 the weakest.


• We can use simplest mutation and crossover methods• By considering a selection method it is easy to calculate the

solution.• Its on your own to calculate the result!

• Have a look at Matlab’s GA toolbox – Just type: gatool

Some Important Applications of GA• Economics

– Biding Strategies, stock trends• Social Systems• Numerical and Combinatorial Optimization

– Job-shop scheduling, Traveling salesman problem(TSP)• Automatic Programming

– Genetic Programming• Machine Learning

– Classification, NN training and designing, • Control

– Gas pipeline, pole balancing, missile evasion• Design Problems

– Semiconductor Design, Aircraft Design, Keyboard configuration, Communication networks, Resource Allocation(e.g. electrical power networks.)

• Robotics:– Trajectory Planning

• Signal Processing: – Filter design

Traveling Salesman Problem(TSP)

• A single salesman travels to each of the cities and completes the route by returning to the city he started from.

• There are cities and given distances between them.• Each city is visited by the salesman exactly once.• Find a sequence of cities with a minimal travelled distance.

Encoding: Chromosome describes the order of cities, in which the salesman will visit them

423835262135327

3846445860697678716967628494

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70 80 90 100

y

x

TSP30 Solution (Performance = 420)

[Genetic Algorithms: A Tutorial: W.Wliliams]

Evolvable Hardware(1)• How to Evolve a Hardware ?! “Design and Optimizing a digital combinational

logic circuit using GA.”• It is important to have a proper encoding.• Assume that your gate are placed on such sheet(Gate-Matrix)

• And each gate is like this(Gate-Characteristic):– Each gate get its input from previous levels: in-1(I,j) in-2(I,j).– Output is connected to a gate: out(I,j).– G(i,j) is position of gate in gate matrix.– Its possible to use several gate types, e.g. : {AND, OR, XOR, NOR, NAND}.

• The only limit is that gate-matrix be a complete set, i.e. it is possible to make any circuit by the use of that set.

• By designing a suitable encoding we can design an algorithm for designing a combinational circuit.

[“Design and Optimizing Digital Combinational Gates”: M.Moosavi, D.Khashabi]

Evolvable Hardware(2)• Fitness Function: What is the best circuit?

– Difference between truth table and output of circuit must be minimum(Zero is desirable)

– Minimum number of gates is desired. – A weighted fitness function:

• N-match is number of output that match truth table.• N-Null is number of Null gates .• W-match is weight(importance) of having true output results.• W-Null is weight(importance) of having minimum gates.

– Most of times W-match/W-null = 10 is a desirable value. • Its important that output be same as truth table even though circuit isn't an

optimized one!–


Evolvable Hardware(3)• An example of results:

– If

– Evolved hardware:

– Fitness-value plot:


Question?Thanks!

App: References:• [1] K.Kiani, Presentation: “Genetic Algorithms” .• [2] A.Dix, M.Sifalakis, Presentation: “Genetic Algorithms”.• [3] W.Wliliams, Presentation: “Genetic Algorithms:A

Tutorial”.• [4] S.N.Razavi, Presentation: “Evolutionary Algorithms”.• [5] E.Hall, Presentation: “Optimization with Genetic

Algorithm/Direct Search Toolbox”• [6] M.Moosavi, D.Khashabi, “Designing and Optimizing

Digital Combinational Logic Circuits”, ISCEE-2010(Submitted!).

Documents

Evolutionary Optimization Wi th focus on genetic algorithm and regarding applications