According to Wikipedia, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints. In particular, a typical problem says maximize $f(x, y)$ subject to $g(x, y) = c$ and follows an equation
$\Lambda (x,y,\lambda) = f(x,y) + \lambda (g(x,y)-c)$
Here's the part I am interested in:
However, 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.
If Lagrangian multipliers aren't a sufficient condition, what is a sufficient condition for optimality in constrained problems?