CONSTRAINST MINIMIZATION SOLVING PROCEDURE

AIM :

To write a equation to solving steps for which calculating an constraint minimization values in given function.

EQUATION SOLVING PROCEDURE :

Function :

A function with respect to x and y.

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

Constraint :

`g(x,y) = x+4y-3`

Convert into lagrangien function :

`l(x,y,lambda) = f(x,y) + lambda(g(x,y))`

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

`therefore (a-b)^2 =a^2+b^2-2ab`

= `5-(x^2+4-4x)-2(y^2+1-2y)+(xlambda+4ylambda+3lambda)`

= `5-x^2-4+4x-2y^2-1+4y+xlambda-4ylambda+3lambda`

Partial differential equation from lagrangien function with respect to `x,y&lambda`

`(deltal)/(deltax)= -2x+4+lambda` ------- eqn 1

`implies-2x+4+lambda = 0 implies 4+lambda =2x implies x= (2+(lambda)/2)`

`(deltal)/(deltay) = -4y+4-4y` -------- eqn 2

`implies 4+4lambda = 4yimpliesy = 1+lambda`

`(deltal)/(deltalambda) =+x+4y-3` -------- eqn 3

Equation 3 is a constraint, we have subsitute the x & y values in equation number 3

`+x+4y-3 =0`

`(2+(lambda)/2)+4(1+lambda)-3 =0`

`lambda = -2/3`

Substitute the `lambda` in x & y expression, we get

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

`y = (1-(2/3)) implies1/3`

RESULTS :

A constraints provide to the given function at which minimize.

x = 5/3, y = 1/3 & `lambda = -2/3`

`f(5/3,1/3) = 4`