Let $A=[a_{ik}]$ be a matrix with the consecutive ones property in each column, i.e. each column consists of a single consecutive block of $1$'s (with zeros everywhere else).  Is there anything at all I can say about the following optimization problem?\begin{align*}
\text{minimize}_{x_{ijk}}\sum_{i}\sum_{j}\sqrt{\sum_{k}c_{k}a_{ik}x_{ijk}} & \,\,\,\,\,\,\,\,\text{subject to}\\
\sum_{j}\sqrt{\sum_{k}c_{k}a_{ik}x_{ijk}} & \leq d_{i}\,\,\forall i\\
\sum_{i}a_{ik}x_{ijk} & =1\,\,\forall j,k\\
x_{ijk} & \geq0
\end{align*}

We assume that $d_i$ and $c_k$ are positive for all $i$ and $k$.  Note that the first constraint contains the same entries as in the objective.