A Crossover for Complex Building Block Overlapping

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A Crossover for Complex Building Block Overlapping

• Miwako Tsuji,
• Masaharu Munetomo,
• Kiyoshi Akama
• (Hokkaido University)

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Problem Definition

• The fitness function f
• Let Vj be the set of loci which consist vj (the
  j-th BB)

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The Existing Crossover Method

• Represent the relation as a graph a BB is a node
  and two overlapped BBs have an edge
• Choose two nodes randomly, then cut the graph
  with minimal edge break (that is, minimize the
  disruption)

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The Partition Tends to HaveSmall Graph

• For a random graph

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The Proposed Method

• Based on the existing method, but refine the
  overlapping graph for pair of parents
• If a BB has identical bits in both parents,
  remove that node
• If the overlapped part of two overlapped BBs have
  identical bits in both parents, remove the edge
• Choose two nodes and minimize the number of cut
  edges

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Illustrating the ProposedCrossover Operator
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Identifying BBs by D5

• Dependency Detection for Distribution Derived
  from df
• How to identify is independent from the crossover
  method
• The algorithm
• For each bit i
• Perturb si and calculate the difference dfi(s)
• Classify the population by dfi(s) into
  sub-populations
• For each sub-population, find the BB Vj by
  minimizing entropy E(Vj)

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Two Phases of D5-GA

• Preprocessing phase
• Detect the possible BBs by D5 algorithm
• Construct those linkage sets from those possible
  BBs
• Evolution phase
• Use the proposed crossover operator, and
• Evolve the population using the linkage
  information from the preprocessing phase
• Two phases runs independently, and may have
  different population sizes

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The Test Function

• Overlapped 5-bit trap function
• The stochastic circularly overlapping
  sub-functions

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The Test Function (cont.)

• The overlapping graph with l 60 and different s

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Experiment Result withKnown Linkage

• The probability of converge (with fixed
  population size)

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Experiment Result withKnown Linkage (cont.)

• The time to convergence (with fixed population
  size)

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Experiment Result withUnknown Linkage

• D5 with the proposed crossover the total number
  of evaluations in the whole phase

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Experiment Result withUnknown Linkage (cont.)

• The proposed crossover the number of evaluations
  only in the evolution phase

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Experiment Result withUnknown Linkage (cont.)

• D5 with the existing crossover the total number
  of evaluations in the whole phase

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Experiment Result withUnknown Linkage (cont.)

• BOA the total number of evaluations

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Conclusions

• A more efficient crossover operator for
  BB-overlapped problems
• Combined the D5 algorithm with the crossover
  operator to solve problems with complexly
  overlapped BBs
• Designed a test function with controllable
  complexity of overlap