e.g. maximize f(x,y)
Subject to g(x,y)=c
We introduce a new variable (λ) called a Lagrange multiplier, and study the Lagrange function defined by
If f(x,y) is a maximum for the original constrained problem, then there exists λ such that (x,y,λ) is a stationary point for the Lagrange function (stationary points are those points where the partial derivatives of Λ are zero). not all stationary points yield a solution of the original problem.
Thus, the method of Lagrange multipliers yields a necessary condition for optimality in constrained problems.
- a powerful tool for solving this class of problems without the need to explicitly solve the conditions and use them to eliminate extra variables.