2) Above we notice that the 2x^2 + 3y^2 <= 100 constraint is not active so we can drop it in which case we can just use upper and lower for those methods that use them.

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.

We can incorporate the constraints within the objective function by returning a large number if any constraint is violated.

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 

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 

1) We can incorporate the constraints within the objective function by returning a large number if any constraint is violated.

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 in which case we can just use upper and lower 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.