Supremum of a function on the open set

Question

I was wandering, if I have a function $F: \mathbb{R}^n\to \mathbb{R}$ and I am loking for its supremum on a certain region $D$ then the standard approach is to look to the points where we have gradient equal to zero, but what if it attains supremum on its boundary. Let us say $D$ is for example a disc, how can one "check" the whole circle for the supremum?

Answer 1

Question has been answered. One way to do that is to use Lagrange Multiplier.

Answer 2

You can formulate your problem as a nonlinear programming, which means you consider the following problem:

$min_x f(x)$ st

$||x||\le \alpha$

In this way if the minimum is attained on the boundaries, the constraint will be active and its corresponding Lagrange multiplier will not be zero. In the other case, it will be zeros and you the problems could be considered as an unconstrained minimization problem.