de {GA} | R Documentation

## Differential Evolution via Genetic Algorithms

### Description

Maximization of a fitness function using Differential Evolution (DE). DE is a population-based evolutionary algorithm for optimisation of fitness functions defined over a continuous parameter space.

### Usage

    de(fitness,
       lower, upper,
       popSize = 10*d,
       stepsize = 0.8,
       pcrossover = 0.5,
       ...)

### Arguments

`fitness` | the fitness function, any allowable R function which takes as input a vector of values representing a potential solution, and returns a numerical value describing its “fitness”.
`lower` | a vector of length equal to the decision variables providing the lower bounds of the search space.
`upper` | a vector of length equal to the decision variables providing the upper bounds of the search space.
`popSize` | the population size. By default is set at 10 times the number of decision variables.
`pcrossover` | the probability of crossover, by default set to 0.5.
`stepsize` | the stepsize or weighting factor. A value in the interval [0,2], by default set to 0.8. If set at `NA` a random value is selected in the interval [0.5, 1.0] (so called dithering).
`...` | additional arguments to be passed to the fitness function. This allows to write fitness functions that keep some variables fixed during the search.

### Details

Differential Evolution (DE) is a stochastic evolutionary algorithm that optimises multidimensional real-valued fitness functions without requiring the optimisation problem to be differentiable.

This implimentation follows the description in Simon (2013; Sec. 12.4, and Fig. 12.12) and uses the functionalities available in the `ga` function for Genetic Algorithms.

The DE selection operator is defined by `gareal_de` with parameters `p = pcrossover` and `F = stepsize`.

### Value

Returns an object of class `de-class`. See `de-class` for a description of available slots information.

### Author(s)

Luca Scrucca luca.scrucca@unipg.it

### References

Scrucca L. (2013). GA: A Package for Genetic Algorithms in R. Journal of Statistical Software, 53(4), 1-37, http://www.jstatsoft.org/v53/i04/.

Simon D. (2013) Evolutionary Optimization Algorithms. John Wiley & Sons.

Price K., Storn R.M., Lampinen J.A. (2005) Differential Evolution: A Practical Approach to Global Optimization. Springer.

### See Also

`summary,de-method`, `plot,de-method`, `de-class`

### Examples

    # 1) one-dimensional function
    f <- function(x)  abs(x)+cos(x)
    curve(f, -20, 20)

    DE <- de(fitness = function(x) -f(x), lower = -20, upper = 20)
    plot(DE)
    summary(DE)

    curve(f, -20, 20, n = 1000)
    abline(v = DE@solution, lty = 3)

    # 2) "Wild" function, global minimum at about -15.81515

    wild <- function(x) 10*sin(0.3*x)*sin(1.3*x^2) + 0.00001*x^4 + 0.2*x + 80
    plot(wild, -50, 50, n = 1000)

    # from help("optim")
    SANN <- optim(50, fn = wild, method = "SANN",
                  control = list(maxit = 20000, temp = 20, parscale = 20))
    unlist(SANN[1:2])

    DE <- de(fitness = function(...) -wild(...), lower = -50, upper = 50)
    plot(DE)
    summary(DE)

    # 3) two-dimensional Rastrigin function

    Rastrigin <- function(x1, x2)
    {
      20 + x1^2 + x2^2 - 10*(cos(2*pi*x1) + cos(2*pi*x2))
    }

    x1 <- x2 <- seq(-5.12, 5.12, by = 0.1)
    f <- outer(x1, x2, Rastrigin)
    persp3D(x1, x2, f, theta = 50, phi = 20, col.palette = bl2gr.colors)

    DE <- de(fitness = function(x) -Rastrigin(x[1], x[2]),
             lower = c(-5.12, -5.12), upper = c(5.12, 5.12),
             popSize = 50)
    plot(DE)
    summary(DE)

    filled.contour(x1, x2, f, color.palette = bl2gr.colors,
                   plot.axes = { axis(1); axis(2);
                                 points(DE@solution,
                                        col = "yellow", pch = 3, lwd = 2) })

    # 4) two-dimensional Ackley function

    Ackley <- function(x1, x2)
    {
      -20*exp(-0.2*sqrt(0.5*(x1^2 + x2^2))) -
      exp(0.5*(cos(2*pi*x1) + cos(2*pi*x2))) + exp(1) + 20
    }

    x1 <- x2 <- seq(-3, 3, by = 0.1)
    f <- outer(x1, x2, Ackley)
    persp3D(x1, x2, f, theta = 50, phi = 20, col.palette = bl2gr.colors)

    DE <- de(fitness = function(x) -Ackley(x[1], x[2]),
             lower = c(-3, -3), upper = c(3, 3),
             stepsize = NA)
    plot(DE)
    summary(DE)

    filled.contour(x1, x2, f, color.palette = bl2gr.colors,
                   plot.axes = { axis(1); axis(2);
                                 points(DE@solution,
                                        col = "yellow", pch = 3, lwd = 2) })

[Package GA version 3.2 Index]