I am trying to formulate the Lagrangian function for the following problem and constraints: $$f(x,y,z)= \sum_{n=1}^N x_{nk}\log(1+y_{nk})+\sum_{m=1}^Mz_{mk}C,$$ and maximize $\sum_{k=1}^Kf(x,y,z)$ subject to$$f_k(x,y,z)\geq b_k,\forall k,\ \sum_{k=1}^K\sum_{n=1}^N y_{nk}\leq Q,\ \sum_{k=1}^K z_{mk}\leq1,\forall m,\ z_{mk}\geq0, \forall k,m. $$
Is the Lagrangian function for the above problem given below correct? \begin{align*} L(x,y,z,\lambda,\nu,\mu_1,\mu_2)&=\sum_{k=1}^Kf(x,y,z)+\sum_{k=1}^K\lambda_k(f_{k}(x,y,z)-b_k)\\ &\mathrel{\phantom{=}} +\nu\left(Q-\sum_{k=1}^K\sum_{n=1}^Ny_{nk}\right)+\sum_{m=1}^M\mu_m\left(1-\sum_{k=1}^Kz_{mk}\right)+\sum_{k=1}^K\sum_{m=1}^M\mu_{mk}z_{mk}. \end{align*}