Retrieving the dual value from a model with quadratic constraints
回答済みHi
I've implemented a LP benders subproblem which fills the strong duality and retrieved the dual values of its constraints properly by calling Pi.
Now I'm extending this benders subproblem to a NLP by adding some quadratic constraints whose fixed model can be linearized at the presolve phase and QCPi are all supposed to be zero. That means the objective of the primal and the dual subproblem shall still be the same after the model is extended.
My approch to retrieve the dual values is to solve the subproblem with NonConvex=2 at first, then I get Pi and QCPi from the linear and the quadratic constraints of the fixed subproblem (with QCPDual=1). However, the values of Pi and QCPi are not but which of RC of the variables are the expected dual values. In addition, it is vice versa if the subproblem is LP (one can get the expected dual value by retrieving Pi but not RC). Could you please explain why this issue happens? Thank you in advance!
Best regards
Zheren
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Gurobi Staff
Hi Zheren,
My approch to retrieve the dual values is to solve the subproblem with NonConvex=2 at first, then I get Pi and QCPi from the linear and the quadratic constraints of the fixed subproblem (with QCPDual=1). However, the values of Pi and QCPi are not but which of RC of the variables are the expected dual values. In addition, it is vice versa if the subproblem is LP (one can get the expected dual value by retrieving Pi but not RC). Could you please explain why this issue happens? Thank you in advance!
I am not sure I fully understand your issue.
You are saying that you have a nonconvex model \(M\) which you solve to optimality to get a solution point \(x^{*}\). You then use some values of \(x^{*}\) to fix nonconvex terms of \(M\) in order to get a convex model \(M^{cv}\). You then solve \(M^{cv}\) with the parameter QCPDual turned on to be able to retrieve the Pi and QCPi attributes. And the issue you see is that these dual values are not correct? Could you please provide a minimal reproducible example for this? Moreover, wouldn't it be better to use \(x^{*}\) in order to fix all nonlinear terms (convex and nonconvex ones) of model \(M\) to make the model an LP instead of a convex nonlinear one?
Best regards,
Jaromił0 -
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Hi Jaromił
You have understand my issue correctly despite my unclear discription.
I've followed the tutorial of the RME and found out that are too many parameters to reproduce. Am I allowed to send you the mps. file of my model?
The model before adding the quadratic constraints was an LP and there was no problem to retrieve the dual values correctly. Therefore it also works properly if I use the solution point of the original problem to fix all nonlinear terms.
Best regards
Zheren
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Hi Zheren,
Uploading files in the Community Forum is not possible but we discuss an alternative in Posting to the Community Forum.
Could you please explain in what terms, the values of Pi and QCPi attributes are incorrect when solving the convex model? In particular, what you think should be the correct values there?
Best regards,
Jaromił0 -
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Hi Jaromił
I've expected that some Pi of the constraints which contain the fixed complicated variable of the original problem are not zero, like the LP. However, after adding the quadratic constraints the values of Pi of those constraints are all zero and thus the density of the benders cut become zero, which signals the improper retrieving.
Best regards
Zheren
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Hi Zheren,
Could you please share an example Model + Script to show the issue? Note that uploading files in the Community Forum is not possible but we discuss an alternative in Posting to the Community Forum.
Best regards,
Jaromił0 -
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Hi
The model of the primal and dual subproblem is based on the paper
"A Benders decomposition approach to transmission expansion planning considering energy storage" without parallel installation and energy storage system."(https://www.sciencedirect.com/sdfe/reader/pii/S0360544216308544/pdf)
without parallel installation and energy storage. The mathmatical formulation of the quadratic constaints added to the primal subproblem is
As I mentioned, they are too many parameters to predefine so that I am not able to make the RME of my model.
Best regards
Zheren
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Hi Zheren,
It is quite hard to help here without any model at hand.
I've expected that some Pi of the constraints which contain the fixed complicated variable of the original problem are not zero, like the LP. However, after adding the quadratic constraints the values of Pi of those constraints are all zero
Gurobi solves an auxiliary KKT problem to compute the dual multipliers. These multipliers are not unique. Thus, I think that the best way for you to proceed here would be to use the solution of the convex model \(M^{cv}\) to fix the nonlinear terms of the model and then solve the LP to retrieve the then correct multipliers for your Benders algorithm.
Best regards,
Jaromił0 -
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Hi Jaromił
Thank you for your tipps. I will follow your suggestion and report the issue or try to implement a convex relaxation like McCormick relaxation if the values of Pi are still unexpected.
Best regards
Zheren
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