Optimize function in R with constraints

Question

I have this function

hazwil2 <- function(diam, flow, leng){
  psi2=((1/(2.31*100))*1050*((flow/140)^1.852)*leng*diam^-4.87)
  return(psi2)
}

I'm trying to use optimize() to minimize psi2 by varying the value of diam. I'm not looking for an absolute minimum but for the smallest value of diam that will keep psi2 below 2. Also, I should mention that diam is the diameter of a pipe, so it can only be a discrete values, 2, 3, 4, 5, or 6.

When I use:

optimize(
  f = hazwil2,
  interval = c(0.1,12),
  flow = 100,
  leng = 400
)

I get the largest possible value of diam (11.9), and a very small value for psi2 . My function is monotonic so the minimization of psi2 will always result in the largest possible diam . I guess I'm struggling to set the constraints for my optimize() function.

Answer 1

uniroot() finds the location where a curve/function crosses zero. Subtract 2 from the PSI to find the minimum allowable diameter. It crosses at 3.564, so 4 is the smallest integer that keeps psi under 2.

intercept   <- 2L
uniroot(
  function(x) hazwil2(x, 100, 400) - intercept ,
  interval = c(3, 4)
)

curve(expr=hazwil2(x, 100, 400), from=2.5, to=5)
abline(a=intercept, b=0, col="blue")

The curve() graph influenced the arbitrary decision to search diameters between 3 and 4.

Result of `uniroot()`:
$`root`
[1] 3.564091

$f.root
[1] -5.618425e-05

$iter
[1] 6

$init.it
[1] NA

$estim.prec
[1] 6.103516e-05