R/goldsteinprice.R

In DiceOptim: Kriging-Based Optimization for Computer Experiments

##' 2D test function
##' 
##' Goldstein-Price 2-dimensional test function.
##' 
##' The goldsteinprice (standardized version) function is defined over the
##' domain \code{[0,1]^2}. It has 1 global minimizer : x* = c(0.5, 0.25), with
##' minimum f(x*) = -3.129172, and 3 local minima x*,2 = c(0.35, 0.4), x*,3 =
##' c(0.95, 0.55), x*,4 = c(0.8 , 0.7), with respective minima f(x*,2) =
##' -2.180396, f(x*,3) = -1.756143, f(x*,4) = -0.807367.
##' 
##' @param x a 2-dimensional vector specifying the location where the function
##' is to be evaluated.
##' @return A real number equal to the goldsteinprice function values at
##' \code{x}
##' @author Tobias Wagner  
##' 
##' Victor Picheny 
##' 
##' David Ginsbourger 
##' @keywords optimize internal
##' @examples
##'  
##' design <- matrix(runif(200), 200, 2)
##' response <- apply(design, 1, goldsteinprice)
##' 
##' @export goldsteinprice
goldsteinprice <- function(x)
{
# Goldstein and Price test function (standardized version)
# --------------------------------------------------------
# Dimension: n = 2
# Number of local minima: 4
# The global minimum: 
# x* = c(0.5, 0.25), f(x*) = -3.129172
# The local minima:
# x*,2 = c(0.35, 0.4),  f(x*,2) = -2.180396
# x*,3 = c(0.95, 0.55), f(x*,3) = -1.756143
# x*,4 = c(0.8 , 0.7),  f(x*,4) = -0.807367

m <- 8.6928
s <- 2.4269

x1 <- 4 * x[1] - 2
x2 <- 4 * x[2] - 2

a <- 1 + (x1+x2+1)^2 * (19 - 14*x1 + 3*x1^2 - 14*x2 + 6*x1*x2 + 3*x2^2)
b <- 30 + (2*x1 - 3*x2)^2 * (18 - 32*x1 + 12*x1^2 + 48*x2 - 36*x1*x2 + 27*x2^2)
f <- log(a*b)
f <- (f-m)/s

return(f)
}