Nonnegative matrix factorization (NMF) has recently been applied to temporal decomposition (TD) of speech spectral envelopes represented by line spectral frequencies. A couple of inherent TD constraints, which are otherwise handled as ad hoc exceptions, has also been incorporated using NMF, including LSF ordering and monotonic event functions. Here, these constraints are analyzed and a third inherent constraint is incorporated into an NMF analysis. This is the complementarity in the sense that two overlapping event functions uniformly add up to one, which has been handled, at best, by a quadratic penalty term. We propose the use of an augmented Lagrangian including a term with Lagrange multipliers (LMs). Additionally, a multiplicative update rule for the LMs is proposed, which fits nicely into the nature of NMF updates. Further, previous difficulties with nonsmooth spectral envelopes have been resolved by obtaining the spectral envelopes from TANDEM-STRAIGHT spectrograms. Good results are reached at the tight event rate of 12.3 ev/s, featuring mean log-spectral distortions ranging from 1.2 dB to about 1.5 dB depending on the regularizations.