Project : Constraint Minimization - Pen and Paper calculation

AIM :

Minimize the following function

`5-(x-2)^2 -2(y-1)^2`

subject to the following constraint

`x + 4y = 3`

Solution :

`f(x,y) = 5 − (x − 2)^2 − 2(y − 1)^2`

`=> 5 − (x^2 + 4 − 4x) − 2(y^2 + 1 − 2y) = 0`

`=> 5 − x^2 - 4 + 4x − 2y^2 - 2 + 4y = 0`

`=> − x^2 − 2y^2 + 4x + 4y - 1 = 0`

By lagrangian method:

`g(x,y) = f(x, y) + λ(x + 4y − 3)`

`g(x,y) = 5 − (x − 2)^2 − 2(y − 1)^2 + λ(x + 4y − 3)`

By equating

`{delg(x,y)}/{delx} = 0` and

`{delg(x,y)}/{dely} = 0`

We get

`{delg(x,y)}/{delx} = - 2 (x-2) + λ = 0 .............(1)`

`{delg(x,y)}/{dely} = - 4 (y-1) + 4λ = 0 ..................(2)`

From equation 1 and 2

`x = {λ+4}/{2} ................... (3)`

`y = λ+1 ................... (4)`

Substituting equation 3 and 4 and constraint equation

`x + 4y = 3`

`=> x + 4y -3 = 0`

`=> ({λ+4}/{2}) + 4(λ+1) -3 = 0`

`=> {λ}/{2}+4/2 + 4λ+4 -3 = 0`

`=> {9λ}/{2}+ 3 = 0`

`=> λ = -2/3`

Put the `λ` value in equation 3 and 4 we get

`x = {-{2}/{3}+4}/{2} = 5/3`

`y = -2/3+1 = 1/3`

Minimum of f(x,y) = `5-(5/3-2)^2 -2(1/3-1)^2 = 36/9 = 4`