We note that optimizing objective (1) subject to constraints (2), (3) and (6) amounts to finding a MSA and can be solved combinatorially (McDonald et al., 2005).
since formulation (1)-(6) is based only on arc variables, by relaxing constraints (4) and 5, one obtains a Lagrangian dual objective which is nothing but a MSA problem with reparameterized arc scores.