_{1}

^{*}

An iterated function system crossover (IFSX) operation for real-coded genetic algorithms (RCGAs) is presented in this paper. Iterated function system (IFS) is one type of fractals that maintains a similarity characteristic. By introducing the IFS into the crossover operation, the RCGA performs better searching solution with a faster convergence in a set of benchmark test functions.

Genetic algorithm (GA) [

Iterated function system (IFS) theory was proposed by Barnsley [

The idea of using IFSX to reproduce offspring is shown in

1) Select 2 parents

2) Combine the information (genes) of

3) Based on the IFS theory [

where

exhibits a self-similarity property. For instance, in _{1}, v_{2} and v_{3}. From (2), we have

9 values_{1}, v_{2} and v_{3}. In some cases, the value of

will generate a random value (within the boundary) to replace it.

4) Randomly pick up n_{para} variables from

then

For example, in

5) Generate the offspring

where Re(Q) and Im(Q) generate vectors formed by the real part and imaginary part of the elements of Q respectively.

Crossover operation is mainly for exchanging information from the two selected parents. In traditional crossover operations (e.g. UNDX and BLX-α), the information is exchanging gene by gene. In the proposed IFSX, each offspring gene is affected by all other genes of the parents, which is a more “complete” crossover operation for information exchange. The IFSX crossover makes the GA operation performs better in terms of fitness value and convergence rate.

The GA with the proposed IFSX goes through six test functions. The results are compared to those from GAs with UNDX and BLX-α. For each test function, the population size is 50 and all the simulation results are averaged ones out of 50 runs. The selection algorithm and the mutation operation are the roulette wheel selection [_{1} is a sphere model which is smooth and symmetric. f_{2} is a step function, which is a representative of flat surfaces. f_{3} is a quartic function which is a simple unimodal function padded with noise. f_{4} is a Shekel’s foxholes function and f_{5} is a Kowalik’s function, which are multimodel functions with only a few local minima. f_{6} is an Ackley’s function which is a multimodel function with many local minima. The parameter λ of the IFSX are set at 0.005, 0.001, 0.001, 0.01, 0.005, 0.005 for f_{1} to f_{6} respectively, and the parameter of the BLX-α is set at 0.336 [

Test functions | Optimal point |
---|---|

IFSX | UNDX | BLX-α | ||
---|---|---|---|---|

f_{1} | Ave. | 2.6783e−19 | 4.9364e−1 | 6.0905e−6 |

S.D. | 9.8771e−19 | 7.6386e−1 | 5.2821e−6 | |

f_{2} | Ave. | 0 | 8.5 | 164.82 |

S.D. | 0 | 5.6973 | 16.241 | |

f_{3} | Ave. | 9.9721e−3 | 1.8537e−1 | 8.9198e−2 |

S.D. | 2.1561e−2 | 1.7033e−1 | 4.1623e−2 | |

f_{4} | Ave. | 0.99942 | 1.0023 | 7.2684 |

S.D. | 0.00688 | 0.02393 | 43.494 | |

f_{5} | Ave. | 5.6569e−4 | 6.1026e−3 | 4.6089e−3 |

S.D. | 7.9694e−4 | 1.0703e−2 | 6.8570e−3 | |

f_{6} | Ave. | 7.444e−12 | 2.8856 | 6.0939e−1 |

S.D. | 2.0494e−11 | 2.4163 | 6.6202e−1 |

In this paper, a new crossover of IFSX for real-coded GA has been proposed. Take the advantage of the iterated function system theory and integrate into crossover operation of real-code genetic algorithm, the solution quality of the searching is enhanced. A suite of benchmark test functions has been used to illustrate the merits of the IFSX.