I am currently trying to use Simulated Annealing package GenSA in order to minimize the function below :
efficientFunction <- function(v) {
t(v) %*% Cov_Mat %*% v
}
Where Cov_Mat is a covariance matrix obtained from 4 assets and v is a weight vector of dimension 4.
I'm trying to solve the Markowitz asset allocation approach this way and I would like to know how I could introduce mathematical constraint such as the sum of all coefficients have to equal 1 :
sum(v) = 1
Moreover since I intend to rely on the GenSA function, I would like to use something like this with the constraint :
v <- c(0.25, 0.25, 0.25, 0.25)
dimension <- 4
lower <- rep(0, dimension)
upper <- rep(1, dimension)
out <- GenSA(v, lower = lower, upper = upper, fn = efficientFunction)
I have found in this paper : http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.97.6091&rep=rep1&type=pdf how to handle such constraint within the Simulated Annealing Algorithm but I don't know how I could implement it in R.
I'd be very grateful for any advice. It is my first time using SO so don't hesitate to tell me if I have the wrong approach in the way I ask question.
A possible approach would be to make use of so-called Lagrange multipliers (cf., http://en.wikipedia.org/wiki/Lagrange_multiplier). For example, set
efficientFunction <- function(v) {
lambda <- 100
t(v) %*% Cov_Mat %*% v + lambda * abs( sum(v) - 1 )
}
, so that in order to minimize the objective function efficientFunction the resulting parameter also minimize the penalty term lambda * abs( sum(v) - 1 ). The Lagrange multiplier lambda is set to an arbitrary but sufficiently high level.
So the function itself doesn't appear to have any constraints that you can set. However, you can reparameterize your function to force the constraint. How about
efficientFunction <- function(v) {
v <- v/sum(v)
t(v) %*% Cov_Mat %*% v
}
Here we normalize the values of v so that they will sum to 1. Then, when we get the output parameters, we need to perform the same transformation
out <- GenSA(v, lower = lower, upper = upper, fn = efficientFunction)
out$par/sum(out$par)
来源：https://stackoverflow.com/questions/23808693/how-to-put-mathematical-constraints-with-gensa-function-in-r