ferpsols: Fitness-Euclidean distance ratio particle swarm optimization...
In ehsan66/emoptim: Evolutionary Multimodal Optimimization

Description Usage Arguments Details Value References Examples

In FER-PSO, Fitness and Euclidean distance Ratio (FER) is calculated based on the fitness difference and the Euclidean distance between a particle's personal best and other personal bests of the particles in the population. The key advantage is that FER-PSO removes the need of prespecifying niching parameters that are commonly required in existing niching evolutiobary algorithms for multimodal optimization.
Furthermore, a local search technique has been used to enhance the ability to locate most global or local optima.

1	ferpsols(fn, lower, upper, control = list(), ...)

`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:

swarm: swarm size. Defaults to 50.
iter: number of iterations. Defaults to 200.
w: inertia weight. Defaults to 0.729843788.
c1: acceleration factor. Defaults to 2.05.
c2: acceleration factor. Defaults to 2.05.
local: logical; local search should be performed? Defaults to TRUE
vectorize_local: logical; vectorization for local search? Defaults to TRUE.
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.

fn is maximized.
This function is efficient when the purpose is finding all global maxima. Please use ncde to find all local maxima.
fn must not return any NaN.
The only stopping rule is the number of iterations.

a list contains:

pbest: a matrix; position of the particles.
pbestval: a vector; corresponding fitness value of each particle.
nfeval: number of function evaluations.
maxima: position of particles after using L-BFGS-B. It should be local maxima. If control$ == FALSE, then it is NA.
maximaval: fitness values of maxima.

Qu, B. Y., Liang, J. J., & Suganthan, P. N. (2012). Niching particle swarm optimization with local search for multi-modal optimization. Information Sciences, 197, 131-143.

Li, X. (2007, July). A multimodal particle swarm optimizer based on fitness Euclidean-distance ratio. In Proceedings of the 9th annual conference on Genetic and evolutionary computation (pp. 78-85). ACM.

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)

ferpsols(two_peak, 0, 20, control = list(seed = 66, swarm = 50))

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

# without refining and local search
ferpsols(two_peak, 0, 20, control = list(seed = 66, swarm = 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 <- ferpsols(dmaxima, 0, 1, control = list(seed = 66, swarm = 100))
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 <- ferpsols(Himmelblau, c(-6, -6), c(6, 6), control = list(seed = 66, swarm = 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 <- ferpsols(Six_Hump, c(-1.9, -1.1), c(1.9, 1.1), control = list(seed = 66, swarm = 200))
unique(round(res$maxima, 5)) ## can not find the local maxima

####################################################################################
## 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 <- ferpsols(Shubert, rep(-10, 2), rep(10, 2), control = list(seed = 66, swarm = 200))
unique(round(res$maxima, 5)) ## only the global maxima are found

## 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)