Constraint Minimization Problem

Aim :

To solve a constraint minimization problem using Lagrange's Multiplier method

Objective :

To minimize the following function using Lagrange multipliers subject to a constraint

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

Constraint : `x + 4y = 3`

Solution :

Given,

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

with respect to `x + 4y - 3 = 0`

Let `lambda`be the Langrange multiplier

=> `g(x,y) = f(x,y) + lambda(constant)`

At minima, `(partialg(x,y))/(partialx) = (partialg(x,y))/(partialy) = (partialg(x,y))/(partiallambda) = 0`

Therefore,

`(partialg(x,y))/(partialx) = -2(x-2) + lambda = 0` ----> (1)

`(partialg(x,y))/(partialy) = -4y(y-1) + 4lambda` ----> (2)

`(partialg(x,y))/(partiallambda) = x +4y-3=0` ----> (3)

From equation (2) & (1) after eliminating `lambda` ,we get `2x - y =3` ----> (4)

Solving equation (3) & (4), we get `x-5y = 0`

Substituting the value of `x` in equation (4), we get the values,

`x =5/3,y=1/3`

Therefore, the constraint is also satisfied with these values

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