Artificial Bee Colony Optimization — abc

Implements Karaboga (2005) Artificial Bee Colony (ABC) Optimization algorithm.

Usage

abc_optim(
  par,
  fn,
  ...,
  FoodNumber = 20,
  lb = rep(-Inf, length(par)),
  ub = rep(+Inf, length(par)),
  limit = 100,
  maxCycle = 1000,
  optiinteger = FALSE,
  criter = 50,
  parscale = rep(1, length(par)),
  fnscale = 1
)

# S3 method for class 'abc_answer'
print(x, ...)

abc_cpp(
  par,
  fn,
  ...,
  FoodNumber = 20,
  lb = rep(-Inf, length(par)),
  ub = rep(+Inf, length(par)),
  limit = 100,
  maxCycle = 1000,
  criter = 50,
  parscale = rep(1, length(par)),
  fnscale = 1
)

# S3 method for class 'abc_answer'
plot(
  x,
  y = NULL,
  main = "Trace of the Objective Function",
  xlab = "Number of iteration",
  ylab = "Value of the objective Function",
  type = "l",
  ...
)

Arguments

par: Numeric vector. Initial values for the parameters to be optimized over
fn: A function to be minimized, with first argument of the vector of parameters over which minimization is to take place. It should return a scalar finite result.
...: In the case of abc_*, further arguments to be passed to 'fn', otherwise, further arguments passed to the method.
FoodNumber: Number of food sources to exploit. Notice that the param NP has been deprecated.
lb, ub: Numeric vectors or scalars. Upper and lower bounds of the parameters to be optimized.
limit: Integer scalar. Limit of a food source.
maxCycle: Integer scalar. Maximum number of iterations.
optiinteger: Logical scalar. Whether to optimize binary parameters or not.
criter: Integer scalar. Stop criteria (numer of unchanged results) until stopping
parscale: Numeric vector of length length(par). Scale applied to the parameters (see stats::optim()).
fnscale: Numeric scalar. Scale applied function. If fnscale < 0, then the problem becomes a maximization problem (see stats::optim()).
x: An object of class abc_answer.
y: Ignored
main, xlab, ylab, type: Passed to graphics::plot().

Value

An list of class abc_answer, holding the following elements:

Foods: Numeric matrix. Last position of the bees.
f: Numeric vector. Value of the function evaluated at each set of Foods.
fitness: Numeric vector. Fitness of each Foods.
trial: Integer vector. Number of trials at each Foods.
value: Numeric scalar. Value of the function evaluated at the optimum.
par: Numeric vector. Optimum found.
counts: Integer scalar. Number of cycles.
hist: Numeric matrix. Trace of the global optimums.

Details

This implementation of the ABC algorithm was developed based on the basic version written in C and published at the algorithm's official website (see references).

abc_optim and abc_cpp are two different implementations of the algorithm, the former using pure R code, and the later using C++, via the Rcpp package. Besides of the output, another important difference between the two implementations is speed, with abc_cpp showing between 50\

Upper and Lower bounds (ub, lb) equal to infinite will be replaced by either .Machine$double.xmax or -.Machine$double.xmax.

lb and ub can be either scalars (assuming that all the parameters share the same boundaries) or vectors (the parameters have different boundaries each other).

The objective function must always return a single finite numeric value. Returning non-finite values (NA, NaN, Inf, -Inf) may cause the optimization to fail.

The plot method shows the trace of the objective function as the algorithm unfolds. The line is merely the result of the objective function evaluated at each point (row) of the hist matrix return by abc_optim/abc_cpp.

For now, the function will return with error if ... was passed to abc_optim/abc_cpp, since those argumens are not stored with the result.

References

D. Karaboga, An Idea based on Honey Bee Swarm for Numerical Optimization, tech. report TR06,Erciyes University, Engineering Faculty, Computer Engineering Department, 2005 https://abc.erciyes.edu.tr/pub/tr06_2005.pdf

Karaboga, D., & Basturk, B. (2007). A powerful and efficient algorithm for numerical function optimization: Artificial bee colony (ABC) algorithm. Journal of Global Optimization, 39(3), 459–471. https://doi.org/10.1007/s10898-007-9149-x

Artificial Bee Colony (ABC) Algorithm (website) https://abc.erciyes.edu.tr/

Basic version of the algorithm implemented in C (ABC's official website) https://abc.erciyes.edu.tr/software.htm

Author

George Vega Yon g.vegayon@gmail.com

Examples


# EXAMPLE 1: The minimum is at (pi,pi) --------------------------------------

fun <- function(x) {
  -cos(x[1])*cos(x[2])*exp(-((x[1] - pi)^2 + (x[2] - pi)^2))
}

abc_optim(rep(0,2), fun, lb=-10, ub=10, criter=50)
#> 
#>  An object of class -abc_answer- (Artificial Bee Colony Optim.):
#>  par:
#>     x[1]:  3.141593
#>     x[2]:  3.141593
#> 
#>  value:
#>           -1.000000
#> 
#>  counts:
#>            186

# This should be equivalent
abc_cpp(rep(0,2), fun, lb=-10, ub=10, criter=50)
#> 
#>  An object of class -abc_answer- (Artificial Bee Colony Optim.):
#>  par:
#>     x[1]:  3.141593
#>     x[2]:  3.141593
#> 
#>  value:
#>           -1.000000
#> 
#>  counts:
#>            232

# We can also turn this into a maximization problem, and get the same
# results 
fun <- function(x) {
  # We've removed the '-' from the equation
  cos(x[1])*cos(x[2])*exp(-((x[1] - pi)^2 + (x[2] - pi)^2))
}

abc_cpp(rep(0,2), fun, lb=-10, ub=10, criter=50, fnscale = -1)
#> 
#>  An object of class -abc_answer- (Artificial Bee Colony Optim.):
#>  par:
#>     x[1]:  3.141593
#>     x[2]:  3.141593
#> 
#>  value:
#>           -1.000000
#> 
#>  counts:
#>            262

# Keep the objective finite over the full search space
fun_safe <- function(x) {
  val <- log(x[1] + x[2])
  if (!is.finite(val)) .Machine$double.xmax else val
}
abc_optim(c(1, 1), fun_safe, lb = -2, ub = 2, criter = 50)
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#> Warning: NaNs produced
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#> Warning: NaNs produced
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#> Warning: NaNs produced
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#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
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#> Warning: NaNs produced
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#> Warning: NaNs produced
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#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
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#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> 
#>  An object of class -abc_answer- (Artificial Bee Colony Optim.):
#>  par:
#>     x[1]: -0.134254
#>     x[2]:  0.135068
#> 
#>  value:
#>           -7.114075
#> 
#>  counts:
#>            71

# EXAMPLE 2: global minimum at about (-15.81515) ----------------------------

fw <- function (x)
  10*sin(0.3*x)*sin(1.3*x^2) + 0.00001*x^4 + 0.2*x+80

ans <- abc_optim(50, fw, lb=-100, ub=100, criter=100)
ans[c("par", "counts", "value")]
#> $par
#> [1] -15.81515
#> 
#> $counts
#> function 
#>      348 
#> 
#> $value
#> [1] 67.46773
#> 


# EXAMPLE 3: 5D sphere, global minimum at about (0,0,0,0,0) -----------------
fs <- function(x) sum(x^2)

ans <- abc_optim(rep(10,5), fs, lb=-100, ub=100, criter=200)
ans[c("par", "counts", "value")]
#> $par
#> [1] -2.989502e-09 -5.469708e-09 -2.061395e-10 -4.430876e-09 -4.197950e-09
#> 
#> $counts
#> function 
#>      449 
#> 
#> $value
#> [1] 7.615277e-17
#> 


# EXAMPLE 4: An Ordinary Linear Regression ----------------------------------

set.seed(1231)
k <- 4
n <- 5e2

# Data generating process
w <- matrix(rnorm(k), ncol=1)     # This are the model parameters
X <- matrix(rnorm(k*n), ncol = k) # This are the controls
y <- X %*% w                      # This is the observed data

# Objective function
fun <- function(x) {
  sum((y - X%*%x)^2)
}

# Running the regression
ans <- abc_optim(rep(0,k), fun, lb = -10000, ub=10000)

# Here are the outcomes: Both columns should be the same
cbind(ans$par, w)
#>             [,1]        [,2]
#> [1,] -0.08051177 -0.08051177
#> [2,]  0.69528553  0.69528553
#> [3,] -1.75956316 -1.75956316
#> [4,]  0.36156427  0.36156427
#             [,1]        [,2]
# [1,] -0.08051177 -0.08051177
# [2,]  0.69528553  0.69528553
# [3,] -1.75956316 -1.75956316
# [4,]  0.36156427  0.36156427


# This is just like OLS, with no constant
coef(lm(y~0+X))
#>          X1          X2          X3          X4 
#> -0.08051177  0.69528553 -1.75956316  0.36156427 
#         X1          X2          X3          X4 
#-0.08051177  0.69528553 -1.75956316  0.36156427