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 get`9y =3`

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

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

as`lamda = (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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