In some applications of convex optimization (e.g. SVM) the way to go is via the Lagrangian Dual Problem.
I'm clear about the following statements (please correct me if I'm wrong):
- The dual problem provides a lower bound for the primal problem.
- The dual problem is always concave. If the primal problem is hard to optimise (not convex, high-dimensional) it's easier to find a lower bound via the dual and use it as an approximation for the primal problem.
- Under KKT-Conditions the optimal solutions of the primal and the dual problem are the same. Therefore the solution can be obtained by both, the primal and the dual problem.
Assuming KKT-conditions, WHEN and WHY is the dual problem the better (or the only) way to go?