Constraint Minimization - Pen and Paper calculation

Objective:

Minimize the following function

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

subject to the following constraint

x + 4y = 3

Solution:

Let  

f(x,y)=5−(x−2)^2−2(y−1)^2

g(x,y)=x+4y−3=0       (1)

 

We solve, ∇f=λ∇g∇f=λ∇g

To simplify, we introduce ‘Lagrangian function’, L and calculate its critical points

L=f−λg  

∴L=5−(x−2)^2−2(y−1)^2+λ(x+4y−3)

 

Thus we solve,

Lx=∂(L/∂)x=−2(x−2)+λ=0         (2)

Ly=∂(L/∂)y=−4(y−1)+4λ=0        (3)

 

from (2)

x=(λ+4)/2      (4)

 

from (3)

y=λ+1             (5)

 

Substituting (4) and (5) in (1)

λ/2+2+4λ+4−3=0

∴9(λ/2)=−3

∴λ=−2/3

 

Substituting (6) in (4) and (5),

We have,

x=5/3;y=1/3

 

Result:

X=5/3

Y=1/3