genetic algorithmsmech.fsv.cvut.cz/~leps/teaching/mom/lectures/lecture06_ga.pdf · genetic...
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Modern optimization methods 1
Genetic Algorithms• Proposed by J. Holland in 60’s during his work
on cellular automata• First „big“ publication for GA function
optimization is a book by D. Goldberg from 80’s• 2015: sear Google for keywords „Genetic
algorithms“ leads to 2 920 000 links (in comparison to „Simulated annealing“1 430 000results)
• As a standard genetic algorithm a binary version of GA from those times is used
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Modern optimization methods 2
Cellular Automata (CA)
Rules:• 3 Black = White• 2 Black = Black• 1 Black = Black• 3 White = White
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Modern optimization methods 3
1D Cellular Automata1-dimensional CA is based on a row of cells. Cells are changing their status/state based on the given rules.
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Modern optimization methods 4
2D Cellular Automata2-dimensional CA is based on finite (infinite) net of cells, where each of them is in one given state. Time is discrete and the state in time t is given by statuses of its neighbors in time t-1.
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Modern optimization methods 5
The universe of the Game of Life is an infinite, two-dimensional orthogonalgrid of square cells, each of which is in one of two possible states, alive or dead.Every cell interacts with its eight neighbours, which are the cells that are horizontally,vertically, or diagonally adjacent. At each step in time, the following transitions occur:
1.Any live cell with fewer than two live neighbors dies, as if by underpopulation.2.Any live cell with two or three live neighbors lives on to the next generation.3.Any live cell with more than three live neighbors dies, as if by overpopulation.4.Any dead cell with exactly three live neighbors becomes a live cell, as if by reproduction.
Example: Conway’s Game of Life
[Wikimedia]
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Modern optimization methods 6
Example: Conway’s Game of Life
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Modern optimization methods 7
.: Conway’s Game of Life
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Modern optimization methods 8
Practical usage: Modeling of concrete microstructural development
W/c = 0.4, RVE 30 x 30 x 30 m
[Vít Šmilauer, Zden k Bittnar]
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Modern optimization methods 9
Example cont.:Development of hydration heat
• SCC, CEM I 42.5 R, 350 kg/m3
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Modern optimization methods 10
Middle Top
Example cont.:Development of hydration heat
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Modern optimization methods 11
Properties of GA• Based on Darwinian idea „survival of the fittest“• Stochastic optimization methods – produce
random behavior• Non-gradient methods – even, no need for
function continuity• Proof of convergence based on so-called Schema
theorem• Usually the methods of the last resort
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Modern optimization methods 12
Survival of the roundest
Gen 1
Selected parents
Gen 2
Gen 3
Selected parents
Winner!
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Modern optimization methods 13
Genetic Algorithms• Notations from biology
x1 = (x11,x1
2,…x1n) f1
x2 = (x21,x2
2,…x2n) f2
|0111011|00001|…|10001|
(solution/individual) (objective/goal function)(variable)
(population)
(coding)
(chromozom)
(gen)
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Modern optimization methods 14
Genetic Algorithms
|0000000000011|
x
9
81
F = 1
F = 2
(genotype) (phenotype) (fitness/force)
E.g.:
DNA Muscle-man Winner of competition
Note: Nowadays the objective function (phenotype) and fitness are usually equal, i.e. f F.
Fast runner
(individual)
3
|0000000001001| 9
f(x) = x2B F = rank
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Modern optimization methods 15
Motivation• Theoretical example:
max f(x) = x2, x[ 0 ; 33 ]
ešeníPo áte ní populace Hodnota x Fitness Pravd pod.
ekávaný po et
Aktuální po et
1 0 1 1 0 1 13 169 0,14 0,58 12 1 1 0 0 0 24 576 0,49 1,97 23 0 1 0 0 0 8 64 0,05 0,22 04 1 0 0 1 1 19 361 0,31 1,23 1
Suma 1170 1 4 4Avg. 292,5 0,25 1 1Max 576 0,49 1,97 2
SolutionInitial Pop. x value Prob.
Exp. count
Actual count
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Modern optimization methods 16
ešeníMating pool
místo ížení Potomci Hodnota x Fitness
1 0 1 1 0 | 1 4 0 1 1 0 0 12 1442 1 1 0 0 | 0 4 1 1 0 0 1 25 6252 1 1 | 0 0 0 2 1 1 0 1 1 27 7294 1 0 | 0 1 1 2 1 0 0 0 0 16 256
Suma 1754Avg. 438,5Max 729
Solution x valueChildrenCross pos.
ešení Potomci Po mutaci Hodnota x Fitness1 0 1 1 0 0 1 1 1 0 0 26 6762 1 1 0 0 1 1 1 0 0 1 25 6252 0 1 0 1 1 0 1 0 1 1 27 7294 1 0 0 0 0 1 0 0 1 0 18 324
Suma 2354Avg. 588,5Max 729
Solution Children x valueAfter mut.
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Modern optimization methods 17
Standard genetic algorithm1 t = 02 Create P0, evaluate P0
3 while (not stopping_criterion) { 4 t = t+15 Choose Mt from Pt-1 (Selection)
6 Alter Mt (Genetic operators)
7 Create Pt from Mt and evaluate Pt (New population)
8 }
Where P is population, or set of possible solutions, M is so-called mating pool, t is number of iterations, here called generation.
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Modern optimization methods 18
Binary coding• Integer numbers are stored in computers in easy
accessible binary format, e.g. using binary operators in C programming language.
• If programming language does not provide tools for binary access, transformation is given by:
1. Length of binary string q
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Modern optimization methods 19
Binary coding
2. Binary string , corresponding to the positive integer value zi, stands
3. With an inversion
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Modern optimization methods 20
Hamming’s barrier
two numbers:
000000010000000 and000000001111111
are as integers neighbors, but in binary code they are very distant(they have big so-called Hamming’s distance)
[Kvasni ka et al., 2000]
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Modern optimization methods 21
Hamming’s barrier: solution
1) Inversion operators
2) Gray coding
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Modern optimization methods 22
Chromosomes’ coding dependent:• For binary coding, Gray’s included:
– One-point crossover
– k-point crossover
111|1111111111
Crossover operators
000|0000000000111|0000000000000|1111111111
parents: children:
111|11111|11111000|00000|00000
111|00000|11111000|11111|00000
parents: children:
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Modern optimization methods 23
Chromosomes’ coding dependent:• For binary coding, Gray’s included:
– Uniform crossover
1111111111111
Crossover operators
000000000000010110001101010100111001010
parents: children:
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Modern optimization methods 24
Chromosomes’ coding dependent:• For real coding of chromosomes, e.g.:
– Arithmetic• Child 1: C1 = * P1 + (1- ) P2
• Child 2: C2 = * P2 + (1- ) P1
is a random number between 0 and 1
– blend crossover BLX-• Two parents P1 and P2 with given bounds L and U• R1 = max (L, P1 (P2 –P1))• R2 = min (U, P2 + (P2 –P1))• Children randomly selected between R1 and R2
usually selected at 0,5
Crossover operators
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Modern optimization methods 25
Special operators:– Partially matched crossover (PMX) – for permutation
tasks
Crossover operators
123|4567|89452|1876|93
xxx|1876|xxxxx|4567|xx
Map:4, 8 5, 7 6, 6 7
x23|1876|x9xx2|4567|93
423|1876|59182|4567|93
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Modern optimization methods 26
Special operators:– Uniform order-based crossover – for permutation
tasks
Crossover operators
123456|789452187|693
123456|xxx452187|xxx
123456|879453287|369..3..6..9
….87.9.Map:
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Modern optimization methods 27
Special operators:– Count preserving crossover (CPC) – ensures the
same amount of 1 and 0
Crossover operators
0101010111001100
x10xx10xx10xx10x
Map:List1 = [4,8]List0 = [1,5]
1100x10x0101x10x
1100010101011100
Rand() = 0.43 < 0.5 exchange
Rand() = 0.78 > 0.5 remain
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Modern optimization methods 28
Mutation operators
123456789 123756489
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Modern optimization methods 29
Selection• Selection is needed to create mating pool, i.e. so-called
parent selection, or to create new generation, i.e. so-called survivor selection
• Dozen of selection methods and tools exist:– Roulette wheel – SUS – Stochastic Universal Sampling– Tournament selection – Inverse tournament selection– Elitism– Ranking a linear scaling
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Modern optimization methods 30
Example
No. Chromosome Value x f(x) Probability of selection
1 01101 13 169 169/1170=0,1442 11000 24 576 576/1170=0,4923 01000 8 64 64/1170=0,0554 10011 19 361 361/1170=0,309Sum 1170 1
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Modern optimization methods 31
Roulette wheel
p1 = 0,14 p2 = 0,49 p4 = 0,31p3 = 0,05
Probability
Cumulativeprobability
0 0,14 0,63 0,68 1
RAND()
RAND()
RAND()
RAND()
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Modern optimization methods 32
Roulette wheel
• Classical tool
• Does not ensure selection of the best solution especially in the case of large populations
• Computationally expensive method, n x RND()
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Modern optimization methods 33
SUS – Stochastic Universal Sampling
c4 = 1,23c3 = 0,22
Expected count
Cumulativeexpected count0 2,55 2,77 4
RAND()
p1 = 0,58 p2 = 1,97
0,58
1,0 1,0 1,0
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Modern optimization methods 34
SUS – Stochastic Universal Sampling
• Does not ensure selection of the best solution especially in the case of large populations, but
• Ensures integer expected counts, e.g.solution with c = 2,0001 will be always selected 2x
• Computationally lest demanding method,only 1 x RND()!
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Modern optimization methods 35
Tournament selection• Randomly selects from population k „contestants“
(most often k = 2) and select the best of them (tournament) and this procedure is done n-times
• Does not ensure selection of the best solution because there is a probability that the best solution would not be selected to the tournament
• Computationally expensive method:k x n x RND()
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Modern optimization methods 36
Inverse tournament selection• Selection of p solutions, where p < n• Randomly selects from population k
„contestants“ (most often k = 2) and the worst one is discarded (inverse tournament) and this procedure is done n-times
• This method ensures selection of the best solution!
• The most „democratic“ method – there exists a probability of survival of the worst solution because of not being selected to the tournament
• Computationally expensive method :k x n x RND() [Hrstka and Ku erová, 2004]
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Modern optimization methods 37
Elitism• Has two senses:1) Selection or copy of the best solution to the next
generation (or mating pool). Used to correct deficiency of selection methods or to control speed of convergence.
2) Property or procedure which preserves actual “better” solutions. For instance, tournament with k = 3 is more elitist than tournament withk = 2.
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Modern optimization methods 38
Ranking and linear scaling
• Methods for control of fitness, supporting or blocking elitism
• Ranking computes fitness not based on objective function value, but on the ranks of solutions within population. E.g. the best solution will have 3 copies, second best 2 copies and reminders to selection one copy. The worst solutions will have 0.
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Modern optimization methods 39
Ranking and linear scaling
• Linear scaling computes fitness to be linearly distributed, e.g. to preserving the average value of objective functions and also ensure the maximal fitness within 1,2-2,0 multiple of average (selected by user)
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Modern optimization methods 40
Proof of convergence: „Schema theorem“two solutions:
01010010100101010101110101010101 and01011010100101110001110111010111
are covered by this schema:
0101#010100101#10#011101#10101#1
Expected count of appearance of this scheme in time t+1is given by
see e.g. [Goldberg, 1989]
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Modern optimization methods 41
• If the whole population is covered by one genotype, the population has converged
• Convergence is premature, if the global optimum has not been found
• Solution:– Higher mutation probabilities– Preserving diversity within population– Multi-modal solutions
Premature Convergence
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Modern optimization methods 42
Deceptive functionsy
0
5
10
15
20
25
30
35
40
45
-8 -6 -4 -2 0 2 4 6 8
Look also for “Robust design optimization”
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Modern optimization methods 43
References
[1] Holland, J. H. (1975). Adaptation in Natural and Artificial Systems. MIT Press.
[2] Goldberg, D. (1989). Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley.
[3] Michalewicz, Z. (1992). Genetic Algorithms + Data Structures = Evolution Programs. AI Series. Springer-Verlag, New York.
[4] Wolfram, S. (2002). Cellular Automata and Complexity. Perseus Publishing.
[5] A. E. Eiben, J. E. Smith (2003). Introduction to Evolutionary Computing. Springer.
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Modern optimization methods 44
References
[6] Dreo J, Petrowsky A, Siarry P and Taillard E. (2006). Metaheuristics for hard optimization. Methods and case studies. Berlin Heidelberg: Springer
[7] V. Kvasni ka, J. Pospíchal, P Ti o (2000). Evolu né algoritmy. STU Bratislava.
[8] Hrstka, O. and Ku erová, A. (2004). Improvements of real coded genetic algorithms based on differential operators preventing the premature convergence. Advances in Engineering Software, 35(3–4):237–246.
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Modern optimization methods 45
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Date of the last version: 12.12.2019Version: 001