# Imposing constraints on observation model in a HMM

I have $$N$$ observations ($$x_1, x_2,.. ,x_N$$) from a HMM with $$K$$ latent states. The M step for computing the observation model $$\mu_k$$ involves maximizing the expression:

$$L = \sum_{n=1}^{N}{ln \sum_{k=1}^K {\pi_k \mathcal{N}(x_n|\mu_k,\Sigma_k) }}$$

Setting $$\frac{\partial L}{\partial \mu_k}$$ to 0, we get

$$\mu_k = \frac{\sum_{n=1}^N \gamma(z_{nk}) x_n}{\sum_{n=1}^N \gamma(z_{nk})} \quad , \text{where } \gamma(z_{nk}) = \frac{\pi_k \mathcal{N}(x_n|\mu_k,\Sigma_k)}{\sum_{j=1}^N \pi_j \mathcal{N}(x_n|\mu_j,\Sigma_j)}$$

(As shown in Bishop page 618)

I am trying to impose constraints on the observation model such that the latent states are ordered according to the mean. (i.e., $$\mu_1 > \mu_2 > \mu_3 ...$$). This will make the problem

$$argmax_{\mu_k} L \quad \text{such that } (\mu_1 - \mu_2) > 0, (\mu_2 - \mu_3 > 0), ...$$

How can I incorporate this constraint to get an expression for $$\mu_k$$?