I am reading a paper that summarizes a set of simulations. Basically, the authors are trying to minimize some function using different optimization algorithms. They conclude: "Our findings point to instances of convergence at points where the first- and second-order optimality conditions fail."
What does this mean in plain English -- that parameter updates have essentially stopped even though the algorithm has not identified an optimal point? My understanding is:
- First-order conditions: Ensure that a critical point has been found -- maximum, minimum, or inflection.
- Second order conditions: Determine that this point is of the nature we want (e.g. that it is indeed a minimum).
Is this correct? My knowledge of optimization is a bit rusty, so I just wanted to confirm that I understand.