Specific Evolutionary Algorithm

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Specific Evolutionary Algorithm

• Part I
• Evolution Strategies

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Outline of an Evolutionary Algorithm (1)

• Definition 2.1 (General Evolutionary Algorithm)
• EA (I, ?, ?, ?, s, ?, ?, ?)
• I Ax ? As is the space of individual.
• ? I ? R denote a fitness function assigning
  real values.
• 
  is a set of
  probabilistic genetic operators, each of which is
  controlled by specific parameters summarized in
  the sets ?i ? R .
• denotes the
  selection operator. An additional set of ?s
  parameters may be used.
• ? the number of parents, ? the number of
  offsprings.
• ? I? ? true, false is a termination criterion
  for the EA.
• ? I? ? I? is the generation transition function.

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Outline of an Evolutionary Algorithm (2)

• Definition 2.2 (Population sequence)
• the sequence P(0), P(1), P(2), is called a
  population sequence or evolution of P(0) ?
  ?t ? 0 P(t1) ?(P(t))
• Definition 2.3 (running time and result of EA)
• the running time ?EA mint?N ?(?t(P(0)))
  true
• the result of an Evolutionary Algorithm

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Outline of an Evolutionary Algorithm (3)

• Definition 2.4 (Asexual, sexual, panmictic
  genetic operators)
• A genetic operators ?? Ip ? Iq is called
• asexual ?
• sexual ?
• panmictic ?

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Outline of an Evolutionary Algorithm (4)

• Algorithm 3 (Outline of an Evolutionary Algorithm)

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Evolution Strategies

• Evolution Strategies are a joint development of
  Bienert, Rechenberg and Schwefel, who did
  preliminary work in this are in the 1960s at the
  Technical University of Berlin.
• Schwefels diploma thesis the discrete mutation
  mechanism ? normally distributed mutation with
  expectation zero and given variance.
  (11)-selection
• Rechenbergs the 1/5-success rule
• A first multimembered ES or (?1)-ES having ? gt 1
  was designed by Rechenberg to inroduce a
  population concept and (?, ?)-ES was introduced
  by Schwefel.
• The (?, ?)-ES characterizes the state-of-art in
  ES research.
• self-adaptation

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• Representation and Fitness Evaluation
• Given an individual
  Rn ? As ,
• n-dimensional object variable vector Rn
• Strategy parameters Each individual may
  additionally include one up to n different
  standard deviations ?i as well as up to n(n-1)/2
  (namely, when n? n) rotation angles ?ij ? -?,
  ? (i ? 1, , n-1, j ? i1, , n)
• The fitness function ? is in principle identical
  to f.

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• Complete description of the generalized
  n-dimensional normal distribution with
  expectation value vector , having probability
  density function.
• C-1 (cij) is the covariance matrix with
  diagonal element cii ?i2, i.e. the variance.
• If covariances do not vanish, the mutation
  ellipsoids may have arbitrary orientation in the
  search space. (n 2, n? 2, n? 1)

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• Because it is not advisable to incorporate the
  covariances directly into the representation and
  apply mutation to them, the rotation angles are
  used.
• The generation of a correlated realization
  from an uncorrelated one can
  be performed by multiplication of ?u by n?
  rotation matrices R(?ij).

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• Mutation
• m?,?,? I? ? I? is an asexual operator.
• n? n, n? n(n-1)/2
• 1 lt n? lt n, n? 0
• n? 1, n? 0

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• A logarithmic normal distribution for the
  variations of standard deviations ?i is motivated
  as follows.
• A multiplicative modification process for ?i
  guarantees positive values of standard
  deviations.
• The median (1/2-quantile) of a multiplicative
  modification must be one (this implies the next
  condition to be fulfilled)
• To guarantee an average neutrality of the process
  in absence of selective pressure, a
  multiplication by a certain value must occur with
  the same probability as multiplication by the
  reciprocal value.
• Smaller modifications must occur more often than
  larger one.

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• Recombination
• , where rx, r? , r? ?-, d, D, i, I, g,
  G, e.g. rdII

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• Theorem 5 (Number of possible recombination
  results)
• Theorem 6 (Generalized discrete recombination)
• The 2nd kind Stirling numbers the number of ways
  of partitioning a set of n elements into i
  nonempty sets
• R(10,2,5) 1.36?103, R(10,2,20) 4.718 ? 107

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• Selection
• The selection operators are completely
  deterministic.
• (??)-selection s(??) I?? ? I?
• In case of changing environments the
  (??)-selection preserves (outdated) solutions
  and is not able to follow the moving optimum.
• (??)-selection hinders the self-adaptation
  mechanism with respect to strategy parameters to
  work effectively, because misadapted strategy
  parameters may survive for a relatively large
  number of generations when they cause a fitness
  improvement by chance.

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• (?,?)-selection s(?,?) I? ? I? (? lt ?)
• The capability of a (?,?)-selection to forget
  good solutions in principle allows for leaving
  (small) local optima and is therefore
  advantageous in case of multimodal topologies.
• A (?,?)-strategy is required in order to
  facilitate extinction of maladapted individuals.
• Selective pressure may not become too strong,
  i.e. ? is required to be clearly larger than one.
• Recombination on strategy parameters in necessary
  (usually, intermediate recombination gives best
  results).
• The ratio ?/? provides the basic parameterization
  instrument for controlling the character of the
  search.

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• Other Components
• creating a complete initial population P(0) from
  a single initial starting point by means of
  mutation
• the initialization of standard deviation ?i(0) ?
  ?xi/n0.5, where ?xi denotes the estimated
  distance between starting point and optimum.
  (3.0)
• Termination is tested by comparing worst and best
  fitness values of the parent population or
  simpler one.

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• Conceptual Standard Algorithm
• Algorithm 4 (Outline of an Evolution Strategy)

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• Definition 2.5 (Evolution Strategy)
• EA (I, ?, ?, ?, s, ?, ?, ?) is called Evolution
  Strategy ?

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• Default hierarchy
• ES(1, 0, rdI, s(15,100))
• ES(n, 0, rdI, s(15,100))
• ES(2, 1, rdII, s(15,100))
• ES(n, n(n-1)/2, rdII, s(15,100))

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• Theory
• 1/5-success rule on average, one out of five
  mutations should cause an improvement with
  respect to objective function values to achieve
  best convergence rates
• The ratio of successful mutations to all
  mutations should be 1/5. If it is greater than
  1/5, increase the standard deviation, if it is
  smaller, decrease the standard deviation.
• Algorithm 5 ((11)-Evolution Strategy)

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• Convergence rate
• the model functions the corridor model, the
  sphere model
• The convergence rate is defined as the
  expectation of the distance k covered by
  mutation, i.e. where p(k)
  denotes the probability for a mutation to cover a
  distance k towards the optimum.

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• The resulting expectation values for the rate of
  convergence, expressed in terms of dimensionless,
  normalized variables ?1 ?1n/b, ?1 ?1n/b,
  ?2 ?2n/r, ?2 ?2n/r (n gtgt 1)
• Expectation values of the corresponding success
  probabilities
  (i ? 1,2 n gtgt 1)

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• By setting it is possible to
  determine the optimal standard deviations that
  maximize the convergence rates and to determine
  the maximum values of convergence rates.
• Standardized convergence rates over standardized
  standard deviation a high robustness of the
  algorithm concerning the choice of ?
• Standardized success probabilities over
  standardized standard deviation optimal values
  are near 0.2
• Whenever the topological situation does not lead
  to success rates larger than 1/5, the search
  stagnates prematurely due to the deterministic
  decrease of step sizes towards zero.

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• To extend the convergence rate theory to cover
  the case of a (1, ?)-strategy with a single
  standard deviation
• The basic idea is to concentrate on the progress
  of the average of the ? best individuals.
• For the restricted case of one single standard
  deviation and absolute coordinates of object
  variable rather than relative ones, the
  probability describing the event that the
  vth-best offspring covers distance k

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• Theorem 8 (Convergence velocity of (?, ?)-ES)

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• The numerical evaluation of the sphere model
  curves
• Convergence velocity may become negative for a
  (1, ?)-ES in case of ?2 becoming too large,
  while the (1?)-strategy guarantees positive
  convergence velocity
• As the number of offspring is increased the
  maximal convergence velocity and the
  corresponding optimal normalized standard
  deviation are also increasing.
• The resulting optimal offspring number yields the
  most reasonable compromise between computational
  effort to be invested and progress rate gain.
  (?1 6, ?2 4.7 ? 5)
• Theoretical results concerning the relation of
  self-adaptation, convergence rate, and ratio of
  parent to offspring number have not been obtained
  yet.

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