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Suppose we have a primal problem $P$ which is stated as a maximization problem $\max c^{T} x$.

The dual problem is defined (Introduction to Linear Optimization by Dimitris Bertsimas) only for a primal minimization problem.

Then what is the dual problem of $P$ ?

Is it implicit, that the dual problem of $P$ is the dual problem of $P$ stated as the minimization problem $\min -c^T x$ ?

Surely these two primal problems are equivalent in the sense that their optimal solution $ \bar x$ are equal (if it exist). However, the objective values are the same only if we ignore the sign !

Shuzheng
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1 Answers1

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I do not have the book of Bertsimas in front of me, but it should state somewhere that "the dual of the dual is again the primal".

So, concerning your question the dual of a $\max$ problem is a $\min$ problem without any need to transform the $\max$ firstly to a $\min$ and then take the dual.

If you insist transforming first to a $\min$ problem and then taking the dual, then it is correct (as you say) that $$\max c^Tx=-\min(-c^T)x$$ so the objective value will be the same but with opposite signs.

Jimmy R.
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  • The book doesn't say what you are saying, however it says: "If we transform the dual into an equivalent minimization problem and then form its dual, we obtain a problem equivalent to the originial problem" - can you say something about this :) ? – Shuzheng Nov 14 '14 at 11:34
  • I found the book online. The dual of the dual is again the primal, that is what you should keep in mind. Wikipedia (http://en.wikipedia.org/wiki/Linear_programming#Duality)starts with a maximization problem as primal. It is the same. Table 4.1 in Bertsimas p. 143 can be used for both directions. You can start from second column and go to the first or from the first and go to the second. – Jimmy R. Nov 14 '14 at 11:46
  • Ahh, under the table 4.1 you are refering too, it says "If we start with a maximazation problem, we can always convert it into an equivalent minimization problem, and then form its dual according to the rules we have just described". I guess, they don't explain the dual of the dual is again the primal, by means of their definition of the dual and to avoid confusion. Do you think I'm right ? Thanks for your help ! – Shuzheng Nov 14 '14 at 11:57
  • Yes, as you say they do not explain it as the dual of the dual, but I think they should have done it, since it is an important simplification and it does not cause confusion! Primal - dual is reflexive (hope this is the correct word) and that causes no confusion. On the contrary their approach with the transformation causes some confusion IMO, but certainly they know much more. I think they refer mainly to engineers and perhaps it serves better their scope that way. Did you check the Wikipedia link to confirm it? – Jimmy R. Nov 14 '14 at 12:02
  • Yep, so in general we can go both ways, that is take the dual of a maximzation problem and obtain the same theory. Can you recommend a better book turned towards mathematicians ? – Shuzheng Nov 14 '14 at 12:04
  • Actually this is a good book. Perhaps that is just a minor "flaw" that you found. Otherwise here is a link with some recommendations http://math.stackexchange.com/questions/20643/linear-programming-books (Bertsimas's book is within the recommendations). – Jimmy R. Nov 14 '14 at 12:07
  • Thank you very much for your help. – Shuzheng Nov 14 '14 at 12:09
  • Hi @Stef I have the same problem. I understand that the dual problem of a maximization primal problem is a minimization problem, however, I am not sure about the constraints of the dual problem. On the table that you mentioned, it is stated that if the variable of the primal is greater or equal to zero, then the constraint of the dual problem will be less than or equal to $c_{i}$ and this case is for the $min$ primal problem. What happen if the primal was a $max$ problem? – FarahFai Dec 05 '14 at 11:11
  • If a variable of the primal is greater or equal to zero, then the constraint of the dual will be in the "right direction" (with respect to the objective function of the dual). The "right direction" or "usual direction" is less or equal for a max objective function and bigger or equal for a min objective function. – Jimmy R. Dec 05 '14 at 11:19