Substituting, we have except the constraint that µ = −c − λ implies that for µ to be positive λ ≤ c. Since this Lagrangian has the same form as equation 3, we can also derive a dual problem, which is the same as in equation 4, with the additional constraint that each λ i can not be bigger than its cost c i .
since we truncate penalties at 0, this suggests that we will learn a penalty of 0 for constraints in three categories: constraints that do not hold in the ground truth, constraints that hold in the ground truth but are satisfied in practice by performing inference in the base CRF model, and constraints that are satisfied in practice as a side-effect of imposing non-zero penalties on some other constraints .