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:

According to Lagrange theory:

For single constraint,the auxiliary function is,

`L(x,y,λ)=f(x,y)-λg(x,y)`

From the given question,

`f(x,y)=5-(x-2)^2-2(y-1)^2`

`g(x,y)=x+4y-3`

Therefore,

`L(x,y,λ)=5-(x-2)^2-2(y-1)^2-λ(x+4y-3)`

`L(x,y,λ)=5-(x-2)^2-2(y-1)^2-λx-4λy+3λ`

Now, we have to partially differentiate L with respect to x , y and λ so as to minimize the given function.

`frac{delL}{delx}=0`

`-2x+4-λ=0 →equation 1`

`x=(-λ)/2+2`

`frac{delL}{dely}=0`

`-y+1-λ=0 →equation 2`

`y=-λ+1`

`frac{delL}{dellamda}=0`

`x+4y=3 →equation 3`

Substituting the values of x and y in equation 3, we get

`x+4y=3`

`(-λ)/2+2+4(-λ+1)=3`

`(-λ)/2+2-4λ+4=3`

`frac{9lambda}{2}=3`

`λ=2/3`

Substituting the value of λ in equation 1, we get

`x=(-2)/6+2`

`x=5/3`

Similarly, substituting the value of λ in equation 2, we get

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

`y=1/3`

Therefore for `x=5/3` ,`y=1/3` and `λ=2/3` we can minimize the function `5-(x-2)^2-2(y-1)^2` subject to the constraint `x+4y-3`