Constraint Minimization

Problem Statement:

Minimize the following function

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

subject to the following constraint

`x + 4y = 3`

Solution:

Given:

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

So we create a new function by adding the existing function to the lagrange multipliers.

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

then substitute the value of f(x,y)

`g(x,y) = 5 – (x-2)^2 -2(y-1)^2 + lambda(x+4y-3)`

differentiate the function w.r.t x to get,

`frac{dg}{dx} = -2(x-2) + lambda = 0 rarr Equation 1`

differentiate the function w.r.t y to get,

`frac{dg}{dy} = -4(y-1) + 4lambda = 0 rarr Equation 2`

differentiate the function w.r.t `lambda` to get,

`frac{dg}{dlambda} = x+4y-3 = 0rarr Equation 3`

Solving the equations 1 and 2 we get

`2x -y -3 =0 rarr Equation 4`

Then solve the equation 3 and 4 to get,

`x = 5y`

substitute the value of x from above equation in the constraint equation to get the value of value of y

`5y + 4y=3`

`9y=3`

`y=1/3`

to obtain x substitute the value of equation `x = 5y`

so the value of `x=5/3`

when we substitute the values of `x=5/3` and `y=1/3` in the constraint equation LHS = RHS so the solution is correct.