How to put mathematical constraints with GenSA function in R

Question

I am currently trying to use Simulated Annealing package GenSA in order to minimize the function below :

efficientFunction <- function(v) {
  t(v)  %*% Co         


        

Answer 1

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)

Answer 2

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.