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Case
1 Suppose you wish to maximize/minimize a function with two
independent
variables,
f(x, y), subject to one constraint,
g(x, y) .
The
functions involved are: f(x, y) =
0 and g(x, y)
= 0
Let f(x,y) be the
function to be optimized subject to the constraint relation, g(x,y).
Both f(x, y) and g(x, y) must be continuously
differentiable functions.
Then
introduce an arbitrary constant, λ ,
called the Lagrange
multiplier such that:
Grad f =
λ Grad g ------------------------------------
( 1 )
where
Grad g means gradient of g
So ∂f/∂x i +
∂f/∂y j =
λ [∂g/∂x i +
∂g/∂y j ]
In
scalar form: ∂f/∂x =
λ [∂g/∂x
], ∂f/∂y =
λ [ ∂g/∂y
], and g(x, y)
= 0
Which
gives 3 equations in the 3 unknowns,
x, y, and
λ
Once
solved, the values of x and
y can be input to f(x, y)
to obtain the optimum.
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