Crossover

In genetic algorithms and evolutionary computation, crossover, also called recombination, is a genetic operator used to combine the genetic information of two parents to generate new offspring.

Recombination Interface

All recombination operations have following call interface: recombination(i1, i2) where i1 and i2 are the same type individuals that involved in recombination to produce an offspring. The recombination function returns pair of recombined individuals.

Note: Some of the selection algorithms implemented as function closures, in order to provide additional parameters for the specified above recombination interface.

Operations

List of the ES strategy recombination operations:

Evolutionary.average — Method

average(ss::Vector{<:AbstractStrategy})

Returns the average value of the mutation parameter $\sigma$ of strategies ss.

List of the ES population recombination operations:

Evolutionary.average — Method

average(population)

Returns an one offspring individual of a multi-parent recombination by averaging population.

Evolutionary.marriage — Function

marriage(population)

Returns an one offspring individual of a multi-parent recombination by random copying from population.

Binary crossovers:

Evolutionary.singlepoint — Function

singlepoint(v1, v2)

Single point crossover between v1 and v2 individuals.

Evolutionary.twopoint — Function

twopoint(v1, v2)

Two point crossover between v1 and v2 individuals.

Evolutionary.uniform — Function

uniform(v1, v2)

Uniform crossover between v1 and v2 individuals.

uniform(r = 1.0)

Returns an in-place real valued mutation function that performs the uniform distributed mutation ^[1].

The mutation operator randomly chooses a number $z$ in from the uniform distribution on the interval $[-r,r]$, the mutation range. The mutated individual is given by

$x_i^\prime = x_i + z_i$

Evolutionary.uniformbin — Function

uniformbin(Cr::Real=0.5)

Returns a uniform (binomial) crossover function, see Recombination Interface, function with the propbabilty Cr ^[2].

The crossover probability value must be in unit interval, $Cr \in [0,1]$.

Evolutionary.exponential — Function

exponential(Cr::Real=0.5)

Returns an exponential crossover function, see Recombination Interface, function with the propbabilty Cr ^[2].

The crossover probability value must be in unit interval, $Cr \in [0,1]$.

Real valued crossovers:

Base.identity — Function

identity(v1, v2)

Returns the same parameter individuals v1 and v2 as an offspring pair.

Evolutionary.discrete — Function

discrete(v1, v2)

Returs a randomly assembled offspring and its inverse from the elements of parents v1 and v2.

Evolutionary.waverage — Function

Weighted arithmetic mean recombination

Evolutionary.intermediate — Function

intermediate(d::Real=0.0)

Returns an extended intermediate recombination operation, see Recombination Interface, which generates offspring u and v as

$u_i = x_i + \alpha_i (y_i - x_i)$
$v_i = y_i + \alpha_i (x_i - y_i)$

where $\alpha_i$ is chosen uniform randomly in the interval $[-d;d+1]$.

Evolutionary.line — Function

line(d::Real=0.0)

Returns a extended line recombination operation, see Recombination Interface, which generates offspring u and v as

$u_i = x_i + \alpha (y_i - x_i)$
$v_i = y_i + \alpha (x_i - y_i)$

where $\alpha$ is chosen uniform randomly in the interval $[-d;d+1]$.

Evolutionary.HX — Function

HX(x, y)

Heuristic crossover (HX) recombination operation^[3] generates offspring u and v as

$u = x + r (x - y)$
$v = y + r (y - x)$

where $r$ is chosen uniform randomly in the interval $[0;1)$.

Evolutionary.LX — Function

LX(μ::Real = 0.0, b::Real = 0.2)

Returns a Laplace crossover (LX) recombination operation^[4], see Recombination Interface.

Evolutionary.MILX — Function

MILX(μ::Real = 0.0, b_real::Real = 0.15, b_int::Real = 0.35)

Returns a mixed integer Laplace crossover (MI-LX) recombination operation^[5], see Recombination Interface.

Permutation crossovers:

Evolutionary.PMX — Function

Partially mapped crossover

Evolutionary.OX1 — Function

Order crossover

Evolutionary.CX — Function

Cycle crossover

Evolutionary.OX2 — Function

Order-based crossover

Evolutionary.POS — Function

Position-based crossover

Tree (expression) crossovers:

Evolutionary.crosstree — Function

crosstree(t1::Expr, t2::Expr)

Perform an arbitrary subtree swap between the expressions t1 and t2.

References

1H. Mühlenbein, D. Schlierkamp-Voosen, "Predictive Models for the Breeder Genetic Algorithm: I. Continuous Parameter Optimization". Evolutionary Computation, 1 (1), pp. 25-49, 1993.
2K. V. Price and R. M. Storn and J. A. Lampinen, "Differential evolution: A practical approach to global optimization", Springer, 2005.
3Z. Michalewicz, T. Logan, S. Swaminathan. "Evolutionary operators for continuous convex parameter spaces." Proceedings of the 3rd Annual conference on Evolutionary Programming, 1994.
4K. Deep, M. Thakur, "A new crossover operator for real coded genetic algorithms", Applied Mathematics and Computation 188, 2007, 895–912
5K. Deep, K. P. Singh, M. L. Kansal, and C. Mohan, "A real coded genetic algorithm for solving integer and mixed integer optimization problems.", Appl. Math. Comput. 212, 505-518, 2009