Is there anything wrong with nlminb in R? - r

Question

I am trying to solve a minimization problem in R with nlminb as part of a statistical problem. However, there is something wrong when comparing the solution provided by nlminb with the plot of the function I am trying to minimize. This is the R-code of the objective function
library(cubature)
Objective_Function <- function(p0){
F2 <- function(x){
u.s2 <- x[1]
u.c0 <- x[2]
u.k0 <- x[3]
s2 <- u.s2^(-1) - 1
c0 <- u.c0^(-1) - 1
k0 <- u.k0/p0
L <- 1/2 * c0 * s2 - 1/c0 * log(1 - k0 * p0)
A <- 1 - pnorm(L, mean = 1, sd = 1)
A <- A * dgamma(k0, shape = 1, rate = 1)
A <- A * dgamma(c0, shape = 1, rate = 1)
A <- A * dgamma(s2, shape = 1, rate = 1)
A * u.s2^(-2) * u.c0^(-2) * 1/p0
}
Pr <- cubature::adaptIntegrate(f = F2,
lowerLimit = rep(0, 3),
upperLimit = rep(1, 3))$integral
A <- 30 * Pr * (p0 - 0.1)
B <- 30 * Pr * (1 - Pr) * (p0 - 0.1)^2
0.4 * B + (1 - 0.4) * (-A)
}
Following the R-command
curve(Objective_Function, 0.1, 4)
one observes a critical point close to 2. However, when one executes
nlminb(start = runif(1, min = 0.1, max = 4),
objective = Objective_Function,
lower = 0.1, upper = 4)$par
the minimum of the function takes place at the point 0.6755844.
I was wondering if you could tell me where my mistake is, please.
Is there any reliable R-command to solve optimization problems?
If this is a very basic question, I apologize.
Thank you for your help.

Answer 1

The problem is not nlminb() but the fact that you have not provided a vectorized function in curve(). You can get the correct figure using the following code, from which you see that nlminb() indeed finds the minimum:
min_par <- nlminb(start = runif(1, min = 0.1, max = 4),
objective = Objective_Function,
lower = 0.1, upper = 4)$par
vec_Objective_Function <- function (x) sapply(x, Objective_Function)
curve(vec_Objective_Function, 0.1, 4)
abline(v = min_par, lty = 2, col = 2)
In addition, for univariate optimization you can also use function optimize(), i.e.,
optimize(Objective_Function, c(0.1, 4))