I know that for general optimization probelm the solution to the dual problem might yeild a lower bound to the primal solution which seems handy.
But, assuming the optimization is $convex$ optimization problem, we know that the solution of the dual problem unites with the primal solution, but in this case of convex optimization problem, it is still required to solve using Lagrange multipliers an optimization problem, so what exactly did we obtain here?
is it the fact that we can use KKT conditions to ease the solution?
is is the fact that I can minimize the lagrangian with respect to $x$ and be sure that I got the solution to the primal?
is it the fact that I can optimize the lagrangian with respect to $\lambda_i$ and plug it back to the lagrangian to optimize with respect to $x$
are those facts that I have written even true?