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From my textbook, I learned bender's decomposition for 2-stage problems and the y variables is non-overlapping. Indeed, I wish to look into this decomposition further since it is very likely a solution to two of my difficult LPs:

https://www.or-exchange.org/questions/10340/method-of-separation-for-lps-with-exponentially-many-constraints https://www.or-exchange.org/questions/10401/simultaneous-column-and-row-generation-reference

but from the textbook, its structural is not exactly what my problem is since its y-variable is not overlapping:

alt text

To address my problem, I hope that there is a version of bender's decomposition with overlapping y variables. Do you have some recommendation (books) that I could look into? Thank you:)

asked 18 Oct '14, 13:59

Chivalry's gravatar image

Chivalry
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accept rate: 0%


Are your \(x\) variables integer-valued? If not, I'm not sure why you would want to use Benders, which is designed for master problems that are MILPs (or, I think, NLPs).

If your mission is just to decompose a large LP into smaller LPs, you might want to look into Dantzig-Wolfe decomposition. If the number of "overlapping" variables between any two blocks is relatively small, you might try the following trick. Suppose that \(y_k\) belongs to two blocks. Create a clone \(y_{k}'\), substitute \(y_{k}'\) for \(y_k\) in one block but not the other, and add a constraint \(y_k = y_{k}'\) to the master problem.

link

answered 18 Oct '14, 14:51

Paul%20Rubin's gravatar image

Paul Rubin ♦♦
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accept rate: 19%

Thank you very much for hinting me on that... I would try that :)

(18 Oct '14, 16:16) Chivalry
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Asked: 18 Oct '14, 13:59

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