I came across the following closed form expression :
$g(\mu, \nu) = \sum_{\mathcal{j}\in\mathcal{U}}\underset{i}{\rm max}\bigl(\text{log}(\epsilon_{ij})-\mu_i\bigr)+\sum_{i\in\mathcal{B}}e^{\mu_i-\nu_iP_i^s-1}+\sum_{i\in \mathcal{B}}v_iP_i^G$ ---(A)
The closed form expression (A) is obtained after substituting eq (1) and eq (2) into eq (B), \begin{equation} g(\mu, \nu)=\begin{cases} \underset{X,K}{\rm max}\sum_{j\in\mathcal{U}}\sum_{i \in \mathcal{B}}x_{ij}\text{log}(\epsilon_{ij})-\sum_{i\in \mathcal{B}}K_i\text{log}(K_i)-\sum_{i\in \mathcal{B}}\mu_i\biggl(\sum_{j\in \mathcal{U}}x_{ij}-K_i\biggr)-\sum_{i\in \mathcal{B}}v_i\bigl(K_iP_i^s-P_i^G\bigr)& \\ \text{such that} \sum_{i \in \mathcal{B}}x_{ij} = 1, x_{ij}\in \{0,1\}, \forall j, \forall i ----(B) \end{cases} \end{equation}
And \begin{equation} x_{ij}^*=\begin{cases} 1, & \text{if $i=i^*$}.\\ 0, & \text{if $i\neq i^*$ }.---(1) \end{cases} \end{equation} Also, $K_i^* = e^{\mu_i-\nu_iP_i^s-1}$ ---(2)
I tried substituting (2) and (1) into (B) but not getting expression (A) properly. Any help in this regard would be highly appreciated.
Note that $\mathcal{U},\mathcal{B}$ are set, $X,K$ are matrix and $\mu, \nu$ are vectors.