Again, the relaxed subproblem may be maximised separately for each block of variables \(x,y,z,\ldots\) In practice, this natural decomposition is usually a bad idea for Lagrangian decomposition: the profit for decision variables \(x\) is too decoupled from the impact of the decision variables \(y, z, \ldots\) in their respective constraint block. Eventually, the pricing problem transfers the costs correctly, but, depending on scaling between the constraints and the objective function, that can take many iterations.
We shouldn't have any scaling issue with a surrogate relaxation. If we apply surrogate relaxation to the previous reformulation, we obtain