I have two sets: $I=\{1,2,\dots,n\}$ and $J=\{1,2,\dots,m\}$ and a set of subsets as $P=\{p_1, p_2, \dots, p_d\}$ that each member of $P$ is a subset of $I$, for example, $p_d=\{3,4\}$. This is my optimization problem where $x_j$ is the decision variable and $z_{ij}$, $LC_i$, $T_{p_d}$, $U_j$ and $c_j$ are the parameters (constants) of the problem:
\begin{align} &\min_{x_j} &\sum_{j \in J} c_j x_j \\ &\text{subject to:} & \sum_{j \in J}\frac{x_j z_{ij}}{x_j - U_j} &\leq LC_i && \forall i \in I\\ &&\sum_{i \in p_d}\sum_{j \in J}\frac{x_jz_{ij}}{x_j - U_j} &\leq T_{p_d} && \forall p_d \in P \\ && x_j &> U_j && \forall j \in J \\ && x_j &\in \mathbb N \end{align}
If I relaxed the problem so that I consider $x_j$ as a continuous variable instead of integer, can I use KKT condition to solve this problem to find a closed form for $x_j$?