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##' 2D test function
##'
##' Branin 2-dimensional test function (standardized version).
##'
##' The branin2 (standardized version) function is defined over the domain
##' \code{[0,1]^2}. It has 3 global minimizers : x*,1 = c(0.1239, 0.8183), x*,2
##' = c(0.5428, 0.1517), x*.3 = c(0.9617, 0.1650), with minimum f(x*,i) =
##' -1.047410
##'
##' @param x a 2-dimensional vector specifying the location where the function
##' is to be evaluated.
##' @return A real number equal to the branin2 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, branin2)
##'
##' @export branin2
branin2 <- function(x)
{
# Branin test function (standardized version)
#--------------------------------------------
# Dimension: n = 2
# Number of local minima: 3 (the global ones)
# The global minima:
# x*,1 = c(0.1239, 0.8183), f(x*,1) = -1.047410
# x*,2 = c(0.5428, 0.1517), f(x*,2) = -1.047410
# x*.3 = c(0.9617, 0.1650), f(x*,3) = -1.047410
m <- 54.8104
s <- 51.9496
xx <- 15 * x[1] - 5
y <- 15 * x[2]
f <- (y - 5.1*xx^2/(4*pi^2) + 5*xx/pi - 6)^2 + 10*(1 - 1/(8*pi))*cos(xx) + 10
f <- (f-m)/s
return(f)
}
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