Mark Miller - 11 months ago 50

R Question

I am attempting to perform constrained optimization in R. I have looked at these posts and a couple of others:

constrained optimization in R

function constrained optimization in R

The first post above is quite helpful, but I am still not obtaining the correct answer to my problem.

My function is:

`Fd <- 224 * d1 + 84 * d2 + d1 * d2 - 2 * d1^2 - d2^2`

and my constraint is:

`3 * d1 + d2 = 280`

First I find the correct answer using an unconstrained exhaustive search followed by a constrained exhaustive search:

`my.data <- expand.grid(x1 = seq(0, 200, 1), x2 = seq(0, 200, 1))`

head(my.data)

dim(my.data)

d1 <- my.data[,1]

d2 <- my.data[,2]

Fd <- 224 * d1 + 84 * d2 + d1 * d2 - 2 * d1^2 - d2^2

new.data <- data.frame(Fd = Fd, d1 = d1, d2 = d2)

head(new.data)

# identify values of d1 and d2 that maximize Fd without the constraint

new.data[new.data$Fd == max(new.data$Fd),]

# **This is the correct answer**

# Fd d1 d2

# 6157 11872 76 80

# Impose constraint

new.data <- new.data[(3 * new.data$d1 + new.data$d2) == 280, ]

# identify values of d1 and d2 that maximize Fd with the constraint

new.data[new.data$Fd == max(new.data$Fd),]

# **This is the correct answer**

# Fd d1 d2

# 14743 11774 69 73

Now find unconstrained maxima using

`optim`

`Fd <- function(betas) {`

b1 = betas[1]

b2 = betas[2]

(224 * b1 + 84 * b2 + b1 * b2 - 2 * b1^2 - b2^2)

}

# unconstrained

optim(c(60, 100), Fd, control=list(fnscale=-1), method = "BFGS", hessian = TRUE)

# $par

# [1] 75.99999 79.99995

Now find constrained maxima using

`constrOptim`

`b1.lower.bound <- c(0, 280)`

b1.upper.bound <- c(93.33333, 0)

b2.lower.bound <- c(93.33333, 0)

b2.upper.bound <- c(0, 280)

theta = c(60,100) # starting values

ui = rbind(c(280,0), c(0,93.33333)) # range of allowable values

theta %*% ui # obtain ci as -1 * theta %*% ui

# [,1] [,2]

# [1,] 16800 9333.333

constrOptim(c(60,100), Fd, NULL, ui = rbind(c(280,0), c(0,93.33333)), ci = c(-16800, -9333.333), control=list(fnscale=-1))

# $par

# [1] 75.99951 80.00798

I have tried playing around with

`ui`

`ci`

`optim`

Thank you for any advice.

Answer Source

`constrOptim()`

uses linear inequality constraints and defines the feasible region by `ui %*% param - ci >= 0`

. If the constraint is `3 * d1 + d2 <= 280`

, `ui`

is `c(-3, -1)`

and `ci`

is `-280`

.

```
Fd <- function(betas) {
b1 = betas[1]
b2 = betas[2]
(224 * b1 + 84 * b2 + b1 * b2 - 2 * b1^2 - b2^2)
}
theta = c(59.999,100) # because of needing `ui %*% inital_par - ci > 0`
ui = c(-3, -1)
ci = -280 # they mean -3*par[1] + -1*par[2] + 280 >= 0
constrOptim(theta, Fd, NULL, ui = ui, ci = ci, control=list(fnscale=-1))
# $par
# [1] 69.00002 72.99993
```

[Edited]
If you want not inequality but equality constraints, it would be better to use `Rsolnp`

or `alabama`

package.

```
library(Rsolnp); library(alabama);
Fd2 <- function(betas) { # -1 * Fd
b1 = betas[1]
b2 = betas[2]
-1 * (224 * b1 + 84 * b2 + b1 * b2 - 2 * b1^2 - b2^2)
}
eqFd <- function(betas) { # the equality constraint
b1 = betas[1]
b2 = betas[2]
(3 * b1 + b2 -280)
}
solnp(pars = c(60, 100), fun = Fd2, eqfun = eqFd, eqB = 0)
auglag(par = c(60, 100), fn = Fd2, heq = eqFd)
```