I have been reading a lot about nonlinear optimization and duality, and it seems that duality theory is extremely useful. I feel that I am missing some of the negative aspects/difficulties associated with it though and am hoping that someone can give me some more perspective.
The fact that the dual function is always concave, for any nonlinear primal optimization problem, seems like quite a significant result. Now I know that the duality gap associated with the optima of the primal and dual may be non-zero, but I feel that I don't have a sense for how large this gap can be for some of the common (nonlinear) problems that arise in optimization.
What are some useful techniques for ensuring that strong duality holds, or being able to say that the duality gap is small enough for practical purposes? Is forming the dual function in attempts to solve a nonlinear program typically a good first-step?
I know this is a rather vague post but I am just hoping to gain some more intuition about the practical usefulness of duality theory in nonlinear optimization.