R help - Sep 2012 - Constraint Optimization with constrOptim

Hi,

I am having trouble using constrOptim. My target is to do a portfolio
optimization and there some constraints have to be fulfilled.

1) The weight of each share of the portfolio has to be greater than 0
2) The sum of these weights has to be 1

I am able to fulfill either the first or the second constraint but not both.

One simple way would be to fulfill the first constraint by using optim as I have
the possibility of using bounds (lower=0, upper=1) but then I am not able to
implement the second constraint.

Has anybody an idea to implement both constraints. Or, does anybody know the
fault in the code (I try to solve it since many days)?

Thank you for your help

Raimund


Here is the simplified code:

#Computation of the minimum-variance portfolio

library(tawny)
library(fBasics)

#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 means
tableMeans <- colMeans(dataset)
  
#Computation of the variance-covariance matrix
covarianceMatrix <- ((t(dataset)) %*% dataset) / 3
  
#--------------------SOLVER-START-----------------------------
  
#start estimates
startingEstimates = rep(1/9, 9)
matrixStartingEstimates = matrix(startingEstimates, nrow=1, ncol=9, byrow=TRUE)

matrixStartingEstimatesNegativ = matrix(rep(-1/9, 9), nrow=1, ncol=9,
byrow=TRUE)
  
#The function which should be minimized
x <- matrixStartingEstimates
fkt <- function(x) {t(x) %*% covarianceMatrix %*% x}
  
#The constraints
#The results should be greater 0                            !DOES NOT WORK!
constraint1 <- x[1:9]
#The summ of the results should be between 0.999 and 1.001
constraint2 <- sum(matrixStartingEstimatesNegativ)
constraint3 <- sum(x)
  
constraintMatrix <- rbind(constraint1, constraint2, constraint3)
constraintVektor <- c(0, -1.00001, 0.99999)
  
#Optimization
constrOptim(startingEstimates, fkt, NULL, ui = constraintMatrix, ci =
constraintVektor)
  

#------------------------SOLVER-END-----------------------------
	[[alternative HTML version deleted]]

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

Dear Ravi,

I wrote r-help at r-project.org some weeks ago because I had troubles doing an
optimization with constraints and your answer was really helpful. Thanks
again. Now I am trying to use spg with another setting: -Inf <= x <= Inf,
sum(x) = 1.

How should proj.simplex look like to do this? I tried many times but I was
not successful.

Best Regards,

Raimund



--
View this message in context:
http://r.789695.n4.nabble.com/Constraint-Optimization-with-constrOptim-tp4643402p4647746.html
Sent from the R help mailing list archive at Nabble.com.