I am trying to solve a non-convex constrained minimization problem. From my understanding, the dual function is always concave and provides a lower-bound to the primal problem for any dual variable(s). Since it is concave, (if it is nice enough) I should be able to solve it using an interior-point algorithm and numerically obtain a lower bound to the primal.
Now, imagine I have found a feasible solution to the primal problem that gives a primal objective value very close to the optimal dual objective value. My questions are:
Does that mean my feasible solution is near optimal? Is this a good way to check if the duality gap is large Or small? or are there more efficient ways to do that?
Why do people try to reformulate or approximate a primal problem with a convex problem even when the duality gap is small enough?