Differential Evolution via Genetic Algorithms
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.
de(fitness, lower, upper, popSize = 10*d, stepsize = 0.8, pcrossover = 0.5, ...)
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 ga function.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.
Returns an object of class de-class. See de-class for a description of available slots information.
Scrucca L. (2013). GA: A Package for Genetic Algorithms in R. Journal of Statistical Software, 53(4), 1-37, tools:::Rd_expr_doi("10.18637/jss.v053.i04") .
Scrucca, L. (2017) On some extensions to GA package: hybrid optimisation, parallelisation and islands evolution. The R Journal, 9/1, 187-206, tools:::Rd_expr_doi("10.32614/RJ-2017-008") .
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.
Luca Scrucca luca.scrucca@unipg.it
summary,de-method, plot,de-method, de-class
# 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) }) # 5) Curve fitting example (see Scrucca JSS 2013) ## Not run: # subset of data from data(trees, package = "spuRs") tree <- data.frame(Age = c(2.44, 12.44, 22.44, 32.44, 42.44, 52.44, 62.44, 72.44, 82.44, 92.44, 102.44, 112.44), Vol = c(2.2, 20, 93, 262, 476, 705, 967, 1203, 1409, 1659, 1898, 2106)) richards <- function(x, theta) { theta[1]*(1 - exp(-theta[2]*x))^theta[3] } fitnessL2 <- function(theta, x, y) { -sum((y - richards(x, theta))^2) } DE <- de(fitness = fitnessL2, x = tree$Age, y = tree$Vol, lower = c(3000, 0, 2), upper = c(4000, 1, 4), popSize = 500, maxiter = 1000, run = 100, names = c("a", "b", "c")) summary(DE) ## End(Not run)