## Constrained Optimization

OBJECTIVE: In this challenge, we will find minimum value of the given function which is constrained by other function.

SOLUTION:

Given:

f(x,y)=5-(x-2)^2-2(y-1)^2 constrained to x+4y=3

Optimization:

g(x,y)=5-(x-2)^2 -2(y-1)^2+lamda(x+4y-3)

(partialg)/(partialx)=0;

implies -2(x-2)+lamda=0----->(A)

(partialg)/(partialy)=0;

implies -4(y-1)+4y =0

implies-(y-1)+y =0-------->(B)

(partialg)/(partiallamda)=0;

impliesx+4y-3=0------>(C)

from equation (A) and(B)

lamda=2(x-2)=(y-1)

implies 2x-y=3----->(D)

from equation (C) AND (D)

x+4y=2x-y

implies x=5y

putting x=5y in equation (C), we get9y =3

impliesy= (1/3)-----(1)

implies x= (5/3)-----(2)

aslamda = (y-1); implies lamda= (-2/3)----->(3)

The minimum value of given function:

f(x,y)=5-(x-2)^2-2(y-1)^2 is at x=(5/3) and y=(1/3)

fmin(x,y)=f(5/3,1/3)=4

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