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Whenever I see articles or books about a $(P)$ optimization problem, they find the dual using the Lagrangian and then call it $$(D)=``\text{Lagrangian dual problem of }(P)"$$ I understand that the purpose of this is to get lower bounds if $(P)$ is maximization problem, so my questions are the following

  1. Are there other ways to get to the same problem $(D)$ without using the Lagrangian?
  2. How else can I construct lower bounds for the optimal value of my problem $(P)$? That is, is there a method that finds a problem $ (D ') $ that gives me better lower limits than $ (D) $?
David Morante
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    You might be interested in the perturbation viewpoint on the dual problem, which I wrote about here: https://math.stackexchange.com/a/624633/40119 – littleO Jul 18 '21 at 06:36
  • thank you very much littleO. – David Morante Jul 18 '21 at 06:39
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    One good thing to be aware of is the Fenchel-Rockafellar approach to duality, where the primal problem is to minimize $f(x) + g(Ax)$ and the dual problem is to minimize $f^(-A^T z) + g^(z)$. Here $f^$ and $g^$ are the convex conjugates of $f$ and $g$, respectively. The Fenchel-Rockafellar dual problem is the same as the Lagrange dual problem you'd obtain by reformulating the primal problem as minimizing $f(x) + g(y)$ subject to $y = Ax$. – littleO Jul 18 '21 at 06:49

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