Hi Raimund,
You can use spg() in the BB package. This requires that you have a projection
rule for projecting an infeasible point onto the simplex defined by: 0 <= x <=
1, sum(x) = 1.
I show you here how to project onto the unit simplex. So, this is how your
problem can be solved:
require(BB)
proj.simplex <- function(y, c=1) {
#####
# project an n-dim vector y to the simplex S_n
# S_n = { x | x \in R^n, 0 <= x <= c, sum(x) = c}
# Copyright: Ravi Varadhan, Johns Hopkins University
# August 8, 2012
#####
n <- length(y)
sy <- sort(y, decreasing=TRUE)
csy <- cumsum(sy)
rho <- max(which(sy > (csy - c)/(1:n)))
theta <- (csy[rho] - c) / rho
return(pmax(0, y - theta))
}
#Rendite - data
dataset <- matrix(c(0.019120, 0.033820, -0.053180, 0.036220, -0.021480,
-0.029380,
-0.012180, -0.076980, -0.060380,
0.038901, -0.032107, -0.034153, -0.031944, 0.006494, -0.021062,
-0.058497, 0.050473, 0.026086,
0.004916, -0.015996, 0.003968, 0.030721, 0.034774, -0.004972,
0.019459, 0.000196, -0.001849), nrow=3, ncol=9, byrow=TRUE)
#Computation of the variance-covariance matrix
covarianceMatrix <- ((t(dataset)) %*% dataset) / 3
fkt <- function(x, covMat) {t(x) %*% covMat %*% x}
startingEstimates = rep(1/9, 9)
ans <- spg(startingEstimates, fn=fkt, project=proj.simplex, covMat =
covarianceMatrix)
ans
sum(ans$par)
ans <- spg(runif(9), fn=fkt, project=proj.simplex, covMat = covarianceMatrix) #
non-uniqueness of the optimal solution
ans
sum(ans$par)
# You see that there is an infinite number of solutiions. Because your
covariance matrix is rank-deficient, it only has rank 3.
# You get a unique solution when the covariance Matrix is positive-definite
aMat <- matrix(rnorm(81), 9, 9)
covMat <- aMat %*% t(aMat)
ans <- spg(runif(9), fn=fkt, project=proj.simplex, covMat=covMat) #
non-uniqueness of the
ans
sum(ans$par)
Hope this is helpful,
Ravi
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