ncde: Neighborhood based crowding Differential Evolution (NCDE) for...
In ehsan66/emoptim: Evolutionary Multimodal Optimimization

Description Usage Arguments Details Value References Examples

A niching differential evolution (DE) algorithm with neighborhood mutation strategy to solve multimodal optimization problems.

1	ncde(fn, lower, upper, control = NULL, ...)

`fn`	objective function that should be maximized. Should not return `NaN`.
`lower`	lower bound.
`upper`	upper bound.
`control`	control parameters for the algorithm. See "Details".
`...`	extra arguments are passed to `fn`.

The control argument is a list that can supply any of the following components:

iter: number of iterations. Defaults to 200.
SF: F is a scale factor in [0, 2] used for scaling the differential vector. Defaults to 0.9.
CR: crossover probability from [0, 1]. Defaults to 0.2.
NP: Population size. Defaults to 10 * length(lower).
seed: random seed.
hybrid: logical; if true, before quiting the algorithm, an L-BFGS-B search with the provided position as initial guess is done to improve the accuracy of the results. Defaults to TRUE. Note that no attempt is done to control the maximal number of function evaluations within the local search step (this can be done separately through hybrid.control)
contr_hybrid: List with any additional control parameters to pass on to stats{optim} when using L-BFGS-B for the local search. Defaults to NULL.

The DE strategy is 'DE/rand/1'.
fn is maximized.
The benchmarks show that ncde is more efficient than ferpsols in finding all local maxima.
fn must not return any NaN.
The only stopping rule is the number of iterations.

The implemented algorithm here slightly differs from the original MATLAB code in a way that the off-springs for all NP members calculated by vectorization (sapply) and not with for loop as the original MATLAB code.

a list contains:

pop: a matrix; position of the population.
popval: a vector; corresponding population values.
nfeval: number of function evaluations.
maxima: position of particles after using L-BFGS-B when control$hybrid = TRUE. It should be local maxima. If control$hybrid == FALSE, then it is NA.
maximaval: fitness values of maxima.

The concept of niching is inspired by the way organisms evolve in nature. When integrated with EAs, niching involves the formation of subpopulations within a population. Each subpopulation aims to locate one optimal solution and together the whole population is expected to locate multiple optimal solutions in a single run (Qu et al., 2012).

Qu, B. Y., Suganthan, P. N., & Liang, J. J. (2012). Differential evolution with neighborhood mutation for multimodal optimization. IEEE transactions on evolutionary computation, 16(5), 601-614.
Based on MATLAB code that can be found in Suganthan's home page.

####################################################################################
## Two-Peak Trap:  global maximum on x = 20 and local maximum on x = 0
two_peak <- function(x)
 y <- (160/15) * (15 - x) * (x < 15) + 40 * (x - 15) * (x >= 15)

ncde(two_peak, 0, 20, control = list(seed = 66, NP = 50))

# without local search
ncde(two_peak, 0, 20, control = list(seed = 66, NP = 50))

# without refining and local search
ncde(two_peak, 0, 20, control = list(seed = 66, NP = 50,  hybrid = FALSE))
## Not run: 
 ####################################################################################
# Decreasing Maxima: one global on x = 0.1 and four local maxima
dmaxima <- function(x)
 y <- exp(-2 * log(2) * ((x - 0.1)/0.8)^2) * (sin(5 * pi * x))^6

res <- ncde(dmaxima, 0, 1, control = list(seed = 66, NP = 10))
unique(round(res$maxima, 5)) ## ferpsols can not find the local maxima

## plot
x <- seq(0, 1, length.out = 400)
plot(x,  dmaxima(x), type = "l")

####################################################################################
# Himmelblau's function: four global optima on
# x = c(-2.80512, 3.13131), c(3.00000, 2.00000), c(-3.77931, -3.28319) and c(3.58443, -1.84813)
Himmelblau <- function(x){
 y <- - (x[1]^2 + x[2] - 11)^2 - (x[1] + x[2]^2 - 7)^2
 return(y)
}


res <- ncde(Himmelblau, c(-6, -6), c(6, 6), control = list(seed = 66, NP = 50))
unique(round(res$maxima, 5))


Himmelblau_plot <- function(x, y)
 Himmelblau(x = c(x, y))
Himmelblau_plot <- Vectorize(Himmelblau_plot)
x <- y <- seq(-6, 6, length.out = 100)
persp(x, y, z = outer(X = x, Y = y, FUN = Himmelblau_plot))

####################################################################################
# Six-Hump Camel Back: two global and two local maxima
Six_Hump <- function(x){
 factor1 <- (4 - 2.1 * (x[1]^2) + (x[1]^4)/3) * (x[1]^2) + x[1] * x[2]
 factor2 <- (-4 + 4 * (x[2]^2)) * (x[2]^2)
 y <- -4 * (factor1 + factor2)
 return(y)
}
res <- ncde(Six_Hump, c(-1.9, -1.1), c(1.9, 1.1), control = list(seed = 66, NP = 10))
unique(round(res$maxima, 5))

####################################################################################
## 2D Inverted Shubert function :
# The global minima: 18 global minima  f(x*) = -186.7309.
# the local maxima: sevral

Shubert <- function(x){
   j <- 1:5
   out <- -(sum(j * cos((j + 1) * x[1] + j)) * sum(j * cos((j + 1) * x[2] + j)))
   return(out)
}

res <- ncde(Shubert, rep(-10, 2), rep(10, 2), control = list(seed = 66, NP = 70))
unique(round(res$maxima, 5)) ## all global maxima but not local maxima

## plotting
Shubert_plot <- function(x, y)
 Shubert(x = c(x, y))
Shubert_plot <- Vectorize(Shubert_plot)
y <- x <- seq(-10, 10, length.out = 40)
persp(x, y, z = outer(X = x, Y = y, FUN =Shubert_plot))

## End(Not run)