Nonlinear constrained optimization with optimx - r

Question

I'm trying to use optimx for a constrained nonlinear problem, but I just can't find an example online that I can adjust (I'm not an R programmer). I found that I should be using the below to test a few algorithms
optimx(par, fn, lower=low, upper=up, method=c("CG", "L-BFGS-B", "spg", "nlm"))
I understand par is just an example of a feasible solution. So, if I have two variables and (0,3) is feasible I can just do par <- c(0,3). If I want to minimise
2x+3y
subject to
2x^2 + 3y^2 <= 100
x<=3
-x<=0
-y<=-3
I guess i can set fn like
fn <- function(x){return 2*x[0]+3*x[1]}
but how do I set lower and upper for my constraints?
Many thanks!

Answer 1

1) We can incorporate the constraints within the objective function by returning a large number if any constraint is violated.
For most methods (but not Nelder Mead) the requirement is that the objective function be continuous and differentiable and requires a starting value in the interior of the feasible region, not the boundary. These requirements are not satisfied for f below but we will try it anyways.
library(optimx)
f <- function(z, x = z[1], y = z[2]) {
if (2*x^2 + 3*y^2 <= 100 && x<=3 && -x<=0 && -y<=-3) 2*x+3*y else 1e10
}
optimx(c(0, 3), f, method = c("Nelder", "CG", "L-BFGS-B", "spg", "nlm"))
## p1 p2 value fevals gevals niter convcode kkt1 kkt2 xtime
## Nelder-Mead 0 3 9 187 NA NA 0 FALSE FALSE 0.00
## CG 0 3 9 41 1 NA 0 FALSE FALSE 0.00
## L-BFGS-B 0 3 9 21 21 NA 52 FALSE FALSE 0.00
## spg 0 3 9 1077 NA 1 0 FALSE FALSE 0.05
## nlm 0 3 9 NA NA 1 0 FALSE FALSE 0.00
1a) This also works with optim where Nelder Mead is the default (or you could try constrOptim which explcitly supports inequality constraints).
optim(c(0, 3), f)
## $par
## [1] 0 3
##
## $value
## [1] 9
##
## $counts
## function gradient
## 187 NA
$convergence
[1] 0
$message
NULL
2) Above we notice that the 2x^2 + 3y^2 <= 100 constraint is not active so we can drop it. Now since the objective function is increasing in both x and y independently it is obvious that we want to set both of them to their lower bounds so c(0, 3) is the answer.
If we want to use optimx anyways then we just use upper= and lower= arguments for those methods that use them.
f2 <- function(z, x = z[1], y = z[2]) 2*x+3*y
optimx(c(0, 3), f2, lower = c(0, 3), upper = c(3, Inf),
method = c("L-BFGS-B", "spg", "nlm"))
## p1 p2 value fevals gevals niter convcode kkt1 kkt2 xtime
## L-BFGS-B 0 3 9 1 1 NA 0 FALSE NA 0.00
## spg 0 3 9 1 NA 0 0 FALSE NA 0.01
## nlminb 0 3 9 1 2 1 0 FALSE NA 0.00
## Warning message:
## In BB::spg(par = par, fn = ufn, gr = ugr, lower = lower, upper = upper, :
## convergence tolerance satisified at intial parameter values.