I'm trying to derive a special form of the Baum Welch algorithm where there is an additional constraint that the sum of emission probabilities over all states sums to one for each output symbol. There is a similar constraint that the sum of emission probabilities for all output symbols sums to one for each state, which is included in the common Baum Welch derivation.

I'm following the derivation here http://ssli.ee.washington.edu/~bilmes/mypubs/bilmes1997-em.pdf on pages 11 and 12, and in particular I'm trying to add my constraint as a Lagrangian multiplier for solving third case, $b_i(k)$. However, when I take the derivative of the function, the two constraints reduce to the sum of both Lagrangian multipliers (since each constraint contains exactly one instance of $b_i(k)$), which seems counterintuitive, as the sum of two Lagrange multipliers doesn't constrain the solution any further than one Lagrange multiplier. Is there a something I'm forgetting?

Thanks in advance for the help.