Constrained optimization of a 2 function of two variable

R Programming
#1

Hello everyone, I have to solve a very difficult problem in R.
I have to maximise the function f(x,y) = x^2 +xy+y^2+3y+2 and I have two constraints:

  • I have to maximise inside the parabola y=x^2-2
  • and I have to respect the constraint y<= 1

I would solve it like this, but I don’t know how to write the matrix A and the vector b.
f ← function (xx,y) xx^2 + xxy +y^2 +3y +2
f

fv ← function (x) f(x[1],x[2])
fv

xx ← seq (-5,5,len=51)
y ← seq (-5,5,len=51)
z = outer (xx,y,f)
image (xx,y,z)
contour (xx,y,z,add=T)

curve (x^2 -2 ,-5,5,add=T)
abline (h=1)

constrOptim (c(0,0),fv,NULL,A,bcontrol=list (fnscale=-1))

I have the exam on Monday, so please help me this weekend.
Thank you in advance,
good by and sorry for my bad english.

#2

Hello Martin,
In R, you can solve this optimization problem using the constrOptim function.

Here’s the code to solve the problem:

f <- function(xx, y) xx^2 + xx * y + y^2 + 3 * y + 2

fv <- function(x) f(x[1], x[2])

xx <- seq(-5, 5, len = 51)
y <- seq(-5, 5, len = 51)

z <- outer(xx, y, f)
image(xx, y, z)
contour(xx, y, z, add = TRUE)

curve(x^2 - 2, -5, 5, add = TRUE)
abline(h = 1)

A <- matrix(c(-1, 2, 0, 1), nrow = 2, byrow = TRUE)
b <- c(-2, 1)

result <- constrOptim(c(0,0), fv, NULL, A, b, control = list(fnscale = -1))

print(result$par)

The optimization problem consists of maximizing the function f(x,y) = x^2 + xy + y^2 + 3y + 2 subject to two constraints: y <= 1 and y = x^2 - 2. The constraints are specified using the matrix A and the vector b. The A matrix has two rows and two columns, where the first row represents the constraint y <= 1 and the second row represents the constraint y = x^2 - 2. The b vector has two elements, where the first element represents the right-hand side of the constraint y <= 1 and the second element represents the right-hand side of the constraint y = x^2 - 2.

The result of the optimization is stored in the result object and can be accessed using the result$par element