I'm searching for the projection from a point to given set, defined by inequalities. Mathematically:
minimize $||X - P||$
whereas x are from:
$A*x <= b$
In my case there are always three variables and arbitrary number of constraints. So matrix A has size n * 3.
To tackle this problem I'm solving the Lagrangian dual problem. Something like https://en.wikipedia.org/wiki/Quadratic_programming#Lagrangian_duality
So far so good, using gradient descent (here it can be more appropriately called 'climb') for maximization of dual function by n variables will end up near solution. There 2 stop criteria I'm using: the length of gradient vector and change in the objective dual function (checking three values -- the ones before and after adding the shift along gradient and one in the middle).
And the question I'm wondering about is -- how can I understand whether the given set is empty and there is no point in searching. Or how such situation affects the gradient iterative method mentioned above.
Thanks in advance.