How to get the maximum likelihood estimate of the categorical distribution parameters using Lagrange optimization?

Let’s say our data is discrete-valued and belongs to one of $K$ classes.
The underlying probability distribution is assumed to be a categorical/multinoulli distribution given as $p(textbf{x}) = prod_{k = 1}^{K}mu_{k}^{x_{k}}$ where x is a one-hot vector given as $textbf{x} = [x_{1} x_{2} … x_{K}]^{T}$ and $boldsymbol{mu} = [mu_{1} … mu_{K} ]^{T}$ are the parameters.
Suppose $D = {mathbf{x}_{1}, text{ } mathbf{x}_{2}, text{ } … ,text{ }mathbf{x}_{N}}$ is our data.

The log likelihood is:
$log p(D|boldsymbol{mu}) = sum_{k = 1}^{K} m_{k} log{mu_{k}}$
where $m_{k} = sum_{n = 1}^{N} x_{nk}$

To get the MLE solution, we have to solve the following optimization problem:
$max_{boldsymbol{mu}} sum_{k = 1}^{K} m_{k} log{mu_{k}} hskip 1em text{such that} hskip 1em mu_{k} geq 0, hskip 0.5em sum_{k = 1}^{K} mu_{k} = 1$

To solve this we write the following Lagrangian.
$L(boldsymbol{mu}, mathbf{u}, v) = sum_{k = 1}^{K} m_{k} log{mu_{k}} – sum_{k = 1}^{K} u_{k}mu_{k} + vleft( sum_{k = 1}^{K}mu_{k} – 1right)$

The primal problem formulation is then
$boldsymbol{hat{mu}} = inf_{boldsymbol{mu}} sup_{u_{k} geq 0, v} L(boldsymbol{mu}, mathbf{u}, v)$

I have no idea how to proceed further. Have no clue how to solve the primal problem.