Optimization in R

Question

I have the following optimization problem:

We are minimizing with respect to y. A is a known matrix, b is a known vector, and c is a known constant.

Two important things here: while we are trying to minimize the function, it can not be less than 0. Also, it would be ideal to use an R method that doesn't require initial values, since it is quite complex to eyeball a feasible starting value of y.

Is there any package/function in R that would allow me to implement this problem? Any help would be greatly appreciated! Thank you.

Answer 1

I second @Stu's suggestion of 'optim'. Here is an quick example of using it for your case:

set.seed(1001)
c = 1
m = 10
b = runif(m)

loss_fun = function(y_i){
  if (all(y_i>=0)){  # <- Here is where we enforce all y's >=0
    return(c - sum(b*log(y_i)))
  }else{
    return(9e9)      # If any y's are negative, return a very large number
  }
}

y_initial = runif(m)

print(optim(y_initial, loss_fun))

Hope that helps.

Update : Sorry confused your 'Ay' for 'For All y'. Try something like this:

set.seed(1001)
c = 1
m = 10
b = runif(m)
A = matrix(runif(m*m), nrow=m)

loss_fun = function(y_i){
  if (all(A%*%y_i>0)){
    return_val = (c - sum(b*log(y_i)))
  }else{
    return_val = 9e9
  }
  if ((c - sum(b*log(y_i)))<0.000001){
    return_val = 9e9
  }
  else{
    return_val = (c - sum(b*log(y_i)))
  }
  return(return_val)
}

y_initial = runif(m)

print(optim(y_initial, loss_fun, method = "L-BFGS-B", lower = 0))

You might have to play around with the different methods and bounds on y to get it to act properly.

The overall idea here is to enforce constraints ('if' conditions in your loss function) to by returning large values if they are broken.

Update 2: If the function returns the bad value '9e9', it will say it converged because it satisfied the loss function. E.g. every other feasible solution gives a value higher than 9e9. This means there is no solution for the method/constraints you have.