Lagrange Multiplier
Lagrange multipliers is a way to solve constrained optimization problems with a form like this:
Minimize (or maximize)
constrained by
It does that by finding critical point at
Find the maximum and minimum values of
on the unit circle Answer: The objective function is
. The constraint is . Lagrange equations: Constraint: The second equation shows
or . Thus, the critical points are
, and .
The Lagrange multipliers doesn’t tell whether a solution is a maximum, minimum, or a saddle point, but we can use the second derivative test.
Intuition
Lagrange multiplier is behind the observation that at maximum or minimum, the level set of function
Proof
Suppose that
Let
Taking a derivative using the chain rule gives
Since
Thus,
However, since