## Constraint Minimization using Lagrangian Method

Minimization of following function with respect to constraint equation

function-  f(x,y)=5-(x-2)^2-2*(y-1)^2

constraint-  lambda(x,y)=x+4y-3

solution:-

According to lagrangian method we consider two equations as one new equation

i.e, g(x,y)=f(x,y) +lambda(x,y)

for minimization partial derivatives = 0

i.e,(partialg)/(partialx)=0

(partialg)/(partialy)=0

(partialg)/(partiallambda)=0

now

(partialg)/(partialx)=partial/(partialx)*(f(x,y)+lambda(x,y))

=partial/(partialx)*(5-(x-2)^2-2*(y-1)^2+x+4y-3)

=-2(x-2)+lambda

=-2x+4+lambda       ------->1

(partialg)/(partialy)=partial/(partialy)*(f(x,y)+lambda(x,y))

=partial/(partialy)*(5-(x-2)^2-2*(y-1)^2+x+4y-3)

=-4(y-1)+4lambda

=-4y+4+4lambda         --------->2

(partialg)/(partiallambda)=partial/(partialx)*(f(x,y)+lambda(x,y))

=partial/(partiallambda)*(5-(x-2)^2-2*(y-1)^2+x+4y-3)

=x+4y-3                  -------->3

solving equations 1 and 2

eq(2)-4*eq(1)

-4y+4+4lambda-4*(-2x+4+lambda)=0

-4y+4+4lambda+8x-16-4lambda=0

8x-4y-12=0                                     ----------->4

solving eq(3) and eq(4)

eq(3)+eq(4)

x+4y-3+8x-4y-12=0

9x-15=0

x=15/9=5/3                                  ----------->5

substituting x value in eq(3)

we get

5/3+4y-3=0

y=1/3

substituting x value in eq(1)

-2*5/3+4+lambda=0

lambda=-2/3

finally

therefore x=5/3,y=1/3,lambda=-2/3

Now

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

substituting the values of x,y

f(x,y)=5-(5/3-2)^2-2*(1/3-1)^2

f(x,y)=4

therefore The minimum value of function f(x,y) is 4.

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