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When solving a MIP with Gurobi, one can use the cbGetNodeRel() function to extract the primal solution to the LP relaxation at each node in the branch and bound tree. Is there similar functionality to extract the dual solution as well? It seems theoretically possible, but I sense that it may be a difficult engineering problem due to Gurobi's extensive pre-solve routine.

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It is not possible, and the reason is that for them to give you the dual, they also need to give you the binding cutting planes at each node. Much of Gurobi's (and CPLEX's as well) magic relies on their proprietary cutting-planes, many of them unknown to us. So you're right, it is possible, but you will never have them give us that info.

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