optim with inequality constraint

Question

I am attempting to use optim() in R to solve for lambda in the following equation:

lambda/sigma^2 - ln(lambda/sigma^2) = 1 + 1/Q

subject to constraint :

lambda > sigma^2.

I am not sure how one goes about setting up this in R.

I am open to alternative optimization routines as well although the equation seems convex and therefore optim should be a fine choice.

Thank you!

Answer 1

You are trying to solve an equation. Whether or not the constraint is met, can only be decided ex post. You can use uniroot as follows

f <- function(x,sigma=1,Q=1) {x/sigma^2 - log(x/sigma^2) - 1 - 1/Q}
uniroot(f,c(1,5))

giving

$root
[1] 3.146198

$f.root
[1] 3.552369e-06

$iter
[1] 5

$estim.prec
[1] 6.103516e-05

Answer 2

Decided this is more an answer than a comment.

Both optim and optimize minimize functions, so what you want to do is write an error function that returns, say, the squared error for a given lambda (se(lambda, sigma^2, Q), make sure your lambda is the first argument). Then call optim(f = se, lower = sigma^2, sigma^2, Q) and it will return the value of lambda that minimizes your error function. If you have multiple data points (Q, sigma^2 pairs) then make your function a sum of squared errors or try using nls().