I want to find a closed-form solution for the following convex optimization problem.
$$ \begin{align} \min_{P_{ij}} & \sum_{i=1}^{N}\sum_{j=1}^{N} \alpha_{ij} P_{ij} \hspace{10ex} \text{ where } \alpha_{ij} \ge 0 \\ \text{s.t.} & \sum_{i=1}^{N}\sum_{j=1}^{N} P_{ij} \log \frac{P_{ij}}{\sum_{j=1}^{N} P_{ij} } + 2\le 0 \\ & \sum_{i=1}^{N}\sum_{j=1}^{N} P_{ij} =1 \\ & 0 \le P_{ij} \le 1 \quad 1 \le i,j\le N \end{align} $$
I have written the KKT conditions, but cannot find my path forward for deriving a closed-form expresion.
Appreciate any tips.