# What does the Dual Function good for in convex optimization other than lower bound?

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?

thanks

• Sometimes the dual is simpler to solve than the primal. Knowing a lower bound is a big deal, it gives some sort of 'guarantee' of how good the current iteration is. – copper.hat Oct 6 '17 at 23:26
• Many optimization algorithms can be interpreted as methods for solving the KKT conditions, which means we are solving the primal and dual problems simultaneously, even if we really only want a solution to the primal problem. By the way, one point to be aware of is that the optimal dual variables (that is, the Lagrange multipliers) have a "sensitvity" interpretation. They tell you (or at least they give you some information about) how much the optimal value to the primal problem will change when the constraints to the primal problem are perturbed slightly. That can be useful sometimes. – littleO Oct 7 '17 at 0:15