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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

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    $\begingroup$ 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. $\endgroup$ – copper.hat Oct 6 '17 at 23:26
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    $\begingroup$ 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. $\endgroup$ – littleO Oct 7 '17 at 0:15
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In many convex optimization problems, we have strong duality. That is, the optimal value of the primal problem is equal to the optimal value of the dual problem. Strong duality also holds for some (but very few) non-convex optimization problems.

Strong duality always holds for Linear Programming problems that have optimal primal and dual solutions. It is also true of more general convex optimization problems that satisfy any of a number of constraint qualifications. For example, Slater's constraint qualification tells us that if a convex optimization problem has a solution that is strictly feasible with respect to its inequality constraints, then strong duality holds.

Many algorithms for convex optimization problems are primal-dual algorithms that simultaneously solve the primal problem and the dual problem. We can measure progress towards finding the optimal solution by verifying that the primal and dual solutions are feasible and checking the gap between the primal and dual objective values of the current solution. We can also use the Karush-Kuhn-Tucker conditions to check the optimality of our final primal and dual solutions.

Compare this with a primal only algorithm. There's no way to tell for sure from a primal solution alone whether or not that solution is optimal. In practice, we often stop when the iterations seem to have stopped changing.

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  • $\begingroup$ In what way the way to solve the primal solution is different than solving for the dual? In both cases I would be needed to solve the same Lagrangian won't I? $\endgroup$ – user3921 Oct 7 '17 at 8:16
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    $\begingroup$ If you're using a primal/dual method, then of course there is no difference, because you're solving both at the same time. But there are other methods that don't have this kind of symmetry, and solve primarily one side of the pair, while still letting you "recover" the other side when you're sufficiently close to the solution. In those cases, it will generally be true that one side is easier to solver than the other. $\endgroup$ – Michael Grant Oct 9 '17 at 1:52

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