CONSTRAINT MINIMIZATION

OBJECTIVE:

The main objective of this challenge is to minimize the given equation according to given constraint

PROBLEM:

`5−(x−2)^2−2(y−1)^2` subject to the following constraint `x+4y=3`

SOLUTION:

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

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

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

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

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

Differentiate w.r.t x and equate to zero

`(delg)/(delx)=-2x+4+lambda=0``rarr`(1)

Differentiate w.r.t y and equate to zero

`(delg)/(dely)=-4y+4+4lambda=-y+1+lambda=0``rarr`(2)

Differentiate w.r.t `lambda` and equate to zero

`(delg)/(dellambda)=x+4y-3=0``rarr`(3)

solve these 3 equations to get values of `x,yand lambda`

`(1)xx1rarr-2x+4+lambda=0`

`(2)xx1rarr-y+1+lambda=0`

solving (1) and (2) we get

`y-2x+3=0``rarr(4)`

(3)`xx2rarr2x+8y-6=0`

`(4)xx2rarr-2x+y+3=0`

solving (3) and (4) we get

`y=1/3`

substitute `y`value in (4) we get `x`

`x=5/3`

substitute `x and y`values in (2) we get `lambda`

`lambda=-2/3`

so final values for `x,y and lambda`

`x=5/3 and y=1/3 and lambda=-2/3`

CONCLUSION:

• Constraint minimization for the given problem is successfully done and calculated the `x,y and lambda values`