Could anyone confirm me if I write the correct dual for my problem? The different sets confuse me a lot.
$s$ is the source node and $t$ the sink node.
I'm uncertain if the last dual constraint in necessary.
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1$\begingroup$ Please use MathJax. You have a typo ($A3$ instead of $A^3$). Also, your primal problem doesn't mention $A^4$ anywhere. And do you have lower bounds $X_a \ge 0$? $\endgroup$– RobPrattMay 16, 2022 at 20:48
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$\begingroup$ Sorry that I missed the $A3$. $A^4$ is part of $A$, but is not in the objective function. It's in the "Notice that". $\endgroup$– orpanterMay 16, 2022 at 20:51
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1$\begingroup$ OK, thank you for the clarification. Two more typos: $x$ should be $X$, and $X_e$ should be $X_a$. $\endgroup$– RobPrattMay 16, 2022 at 21:03
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1 Answer
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Looks almost correct to me. Two issues:
- Your $ \le$ dual constraints suggest that you omitted $X_a \ge 0$ in the primal. If instead $X_a$ is free, then your dual constraints should be $=$.
- Your Tail and Head are reversed in the dual.
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$\begingroup$ Thank you so much! I spend two whole week fighting with this! My first time with a model like this with so many subsets. $\endgroup$– orpanterMay 16, 2022 at 21:06
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$\begingroup$ Glad to help. Please mark my answer as accepted. Also, you might find Arthur Benjamin's S-O-B Method useful. $\endgroup$– RobPrattMay 16, 2022 at 22:28