Constraint Minimization

Objective:

To minimize the function `f(x,y) = 5-(x-2)^2-2(y-1)^2` subject to the constraint `x+4y=3` using Lagrange Multipliers.

Solution:

Given equation:

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

Simplifying the equation

`f(x,y) = 5-(x^2-4x+4)-2(y^2-2y+1)`

`f(x,y) = -x^2-2y^2+4x+4y-1` .(I)

Constraint minimization using Lagrange Multiplier:

New equation

`g(x,y)=f(x,y)+lamda(x+4y-3)` .(II)

Substituting eqn. I in II

`g(x,y)=-x^2-2y^2+4x+4y-1+lamda(x+4y-3)``

Differential Equations `(delg)/(delx)=0, (delg)/(dely)=0, (delg)/(dellamda)=0`

Differentiating eqn III wrto x

`(delg)/(delx)=-2x+4+lamda` .(IV)

Differentiating eqn III wrto y

`(delg)/(dely)=-4y+4+4.lamda` .(V)

Differentiating eqn III wrto `lamda`

`(delg)/(dellamda)=x+4y-3` .(VI)

From eqn IV

`(delg)/(delx)=0`

`-2x+4+lamda = 0`

`x=(4+lamda)/2` .(VII)

From eqn V

`(delg)/(dely)=0`

`-4y+4+4.lamda=0`

`y = 1+lamda`

Substitute the value of x and y in eqn VI

`(delg)/(dellamda)=0`

`x+4y-3 = 0`

`(4+lamda)/2+4(1+lamda)-3=0`

`2+(lamda)/2+4+4lamda-3=0`

`(9lamda)/2+3=0`

`lamda=-6/9`

`lamda=-2/3`

Substitute the value in eqn VII and VIII

`x=(4+lamda)/2`

`x=(4+(-2/3))/2`

`x=10/6`

`x=5/3`

`y = 1+lamda`

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

`y = 1/3`

Result:

From solving the above equations, calculated the values of `x=5/3`, `y = 1/3` and `lamda=-2/3` which minimizes the equation `f(x,y) = 5-(x-2)^2-2(y-1)^2` subject to the constraint `x+4y=3`.