010'141 Engineering Mathematics II Lecture 13 Evolutionary Algorithms

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010.141 Engineering Mathematics IILecture
13Evolutionary Algorithms

• Bob McKay
• School of Computer Science and Engineering
• College of Engineering
• Seoul National University

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Outline

• Feedback
• Simple example of applying an evolutionary
  algorithm
• Variants of the classical evolutionary algorithm
• Realistic evolutionary algorithm example

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Reminder Evolutionary Algorithm

• Pop(0)random t0
• While not (solution ? Pop(t)) do
• while size(Pop(t1)) lt size(Pop(t)) do
• randomly select Ind1, Ind2 ? Pop(t)
• with probability p1
• Newrandom_mutate(Ind1)
• else with probability p2
• Newrandom_combine(Ind1, Ind2)
• else New Ind1
• add New to Pop(t1)
• end while
• tt1
• End while

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Genetic Algorithm Example

• We will use the simple GA to maximise the
  function f(x) x2, with x in the integer
  interval 0 31, i.e., x 0 1 30 31
• Of course, the maximum at x31, but pretend we
  dont know this
• The first step of EA applications is encoding
• i.e., the representation of chromosomes
• We adopt binary representation for integers.
• Five bits are necessary for integers up to 31
• Suppose the population size is 4

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Generate initial population at random

• e.g., 01101, 11000, 01000, 10011

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Calculate fitness value for each individual

• Decode the individual into an integer,
• 01101 ?13 11000 ?24 01000 ?8 10011 ?19
• Evaluate the fitness according to f (x) x2,
• 13 ?169 24 ?576 8 ?64 19 ?361

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Selection Probabilities

• p1(13) 169/1170 0.14
• p2(24) 576/1170 0.49
• p3(8) 64/1170 0.06
• p4(19) 361/1170 0.31

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Select individuals for crossover

• Select two individuals (with replacement)
  according to the above probabilities for
  crossover
• Say we have crossover(01101 11000) and
  crossover(10011 11000)
• We obtain offspring 01100 and 11001 from
  crossover(01101 11000) by choosing a random
  crossover point at 4
• We obtain 10000 and 11011 from crossover(10011
  11000) by choosing a random crossover point at 2
• Now we have 01100 11001 10000 11011

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Select individuals for mutation

• in the newly formed (intermediate) population
  with a small probability
• i.e., randomly change 0(1) to 1(0) in 01100
  11001 10000 11011
• Now we have the new population P (1) 01101
  11001 00000 11011

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Repeat if the population has not converged.

• P (1) 01101 11001 00000 11011
• P (2) 00001 11101 00011 11010
• P (3) 00001 11111 11110 11001
• P (4) 11111 11110 11101 11101
• P (5) 11111 11101 11111 11110
• Etc

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Remarks

• There is no restriction on the fitness or
  objective function
• It can be non-differentiable or even
  discontinuous
• There is no need to know the exact form of the
  objective function
• If the objective function is too complex to
  express explicitly, it can still be simulated
  since EAs only use function values
• In fact, more modern selection algorithms just
  use solution ordering, so objective values just
  have to be orderable
• There can be many different genetic operators and
  selection mechanisms
• The initial population does not have to be
  generated at random.
• The representation of chromosomes does not have
  to be binary.
• Domain knowledge can be incorporated into
  representation and genetic operators.

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Variants Crossover

• One-point crossover
• 011011 ? 011101
• 101101 101011
• k-point crossover (k gt 1)
• xxxxxxxxxx ? xxoooxxxxo
• oooooooooo ooxxxoooox
• Uniform crossover
• xxxxxxxxxx ? xooxxoxoxx
• 1001101011
• oooooooooo oxxooxoxoo

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Variants Crossover

• Intermediate crossover (
• for continuous representation of chromosomes only
• xoffspring x2 r(x1 - x2)
• where r is a uniformly distributed random number
  in 0 1
• Other crossover, e.g., order-based crossover

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Variants Mutation

• Mutation on binary strings (bit-flipping)
• Mutation on real values (Gaussian mutation)
• Adaptive Mutation
• Gaussian mutation with varying ?
• or ? may be part of the chromosome
• Problem-specific mutation

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Variants Selection

• Truncation selection
• Select the k fittest individuals
• Roulette wheel selection
• Fitness-proportionate selection
• Likelihood of selection is proportional to
  fitness
• Rank-based selection
• Likelihood of selection is proportionate to rank
• Tournament selection (size k)
• Randomly choose k individuals
• Select the fittest of them

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Variants other mutation operators

• Problem-specific operators
• You can design operators to match the problem
  representation
• Biologically inspired operators
• Insertion
• Deletion
• Replication
• Transposition
• Correspond to the real mutations that occur in
  DNA

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Variants other recombination operators

• Differential evolution
• A kind of 3-way crossover given x1, x2, x3
• xnew x3 r(x1 - x2)
• Multi-parent crossover
• given x1, x2, , xr
• xnew a1x1 a2x2 arxr
• a1, a2, , ar ? -?, 1 ?
• ? usually chosen as 0.5

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Travelling Salesman Problem (TSP)

• Find the (a) shortest tour of N cities visiting
  each city exactly once and returning to the
  starting city
• Or expressed mathematically
• Given N cities, 1, 2,..., n, and distances
  dij, 1 i, j N, among them
• Find a permutation (x1, x2,..., xN ) of (1,
  2,..., N )
• Such that D ?I1N dxixi1 is minimum
• (where xN1 x1)
• An NP-Complete Problem

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TSP Representation

• Represent each tour as a tuple
• (x1, x2,..., xN) means visiting city x1 first,
  then x2, then ..., , then xN, and finally going
  back to x1

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TSP Mutation

• Reverse any segment of the tour
• Similar to Lin's 2-opt neighbourhood system

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TSP Recombination

• Construct an edge map from 2 parental tours

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TSP Recombination

• Construct a child tour from the edge map
• Choose the initial city at random as the current
  city
• Determine which of the cities in the edge list of
  the current city has the fewest entries in its
  own edge list
• City with the fewest entries becomes current city

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Some Variants Selection

• Fitness can be defined as inverse of tour length
• f 1 / D
• Roulette Wheel Selection
• Let fitness of solution j in a population, G(j),
  be fj, 1 j M
• Then solution j's reproduction probability is
• PGA5 (j) fj / ?k1M fk

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Some Variants Selection

• Rank-Based Selection
• Tours in a population are first sorted in a
  non-descending order according to their lengths
• Let the M sorted tours be numbered as 0, 1,...,
  M-1
• Then the Mjth tour is selected with probability
• PGA6 (M - j) j / ?k1M k
• Competition
• The probability of attaining a win over the
  opponent is the opponent's fitness divided by the
  sum of the two competing solutions' fitness
• For example, if solution i's fitness is 100 and
  solution j's fitness is 200, then solution i wins
  with probability 2/3

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We look at six Approaches

• GA1, GA2, and GA3 use recombination only, no
  mutation
• GA4, GA5, and GA6 use mutation only, no
  recombination
• GA1 Recombination as described above
• GA2 Favour the nearest neighbour
• GA3 Tie is broken in favour of a nearer
  neighbor
• GA4 Competition
• GA5 Roulette wheel selection
• GA6 Rank-based selection

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Experiments

• We compare GA1 to GA6 on a TSP
• Ten randomly generated TSPs with 100 cities
• The expected minimum tour length is 100
• Performance Metrics
• Best (found so far)
• Mean (in a population)
• Entropy (in a population)
• H 1 / N ?i1N H i
• where N is the number of cities and
• H i -1 / log(N) ?j1N (nij / 2P) log (nij /
  2P)
• P is the population size, nij is the number of
  edges connecting city i and city j in the
  population

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ResultsBest Tour Length vs of generations
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ResultsMean Tour Lgth vs of generations
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ResultsEntropy vs of generations
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Analysis of Results

• GA2 looks best
• But is it
• If we look early, GA3 looks better
• But it stops searching effectively
• Because it loses diversity
• Crossover becomes ineffective
• Notice GA1, 4, 5, 6 retain more diversity
• Maybe they will get better if we search longer

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Exploration vs Exploitation

• An evolutionary algorithm has a trade-off between
• Exploitation
• Looking very carefully near already good points
• Hill-climbing is the extreme case
• Exploration
• Looking very widely at good points
• Random search is the extreme case

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Exploration vs Exploitation

• The right trade-off is problem-specific
• If the fitness landscape is very smooth
• Find the maximum of f(x)x2
• Then the algorithm should emphasise exploitation
• Stochastic hillclimbing will out-perform any
  evolutionary algorithm
• Deterministic hillclimbing (gradient descent)
  will do even better

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Exploration vs Exploitation

• If the fitness landscape is very rough
• Find good paths in TSP
• Then the algorithm should emphasise exploration
• Stochastic hillclimbing will perform very poorly
• Methods to increase diversity (fitness sharing)
  may be very important

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Fitness Landscapes

• Sometimes we can know about the fitness landscape
  before we start
• More commonly, we have to discover about the
  fitness landscape
• By experimenting with the performance of
  algorithms
• By directly measuring
• Fitness - distance correlation
• The better the correlation, the smoother the
  landscape

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Summary

• Simple example of applying an evolutionary
  algorithm
• Variants of the classical evolutionary algorithm
• Realistic evolutionary algorithm example

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