Nonlinear constrained optimization with optimx-CodePudding

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!

CodePudding user response：

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.