Given a set $j \in J$, a set $i \in I$, positive coefficients $C_j$, positive values $w_{ij}$. For each $j$ find set of values of $i$, $I_j$, such that $\sum_{j \in J} C_j (1- \sum_{i \in I_j}w_{ij})^2$ is minimized subject to the constraints $\sum_{i \in I_j}w_{ij} \le 1$, $\cup_j I_j = I$ and $I_j \cap I_j'= \phi, \forall j,j' \in J$. Or, find a disjoint partitioning of the set $I$ such that the given function is minimized subject to the less than 1 constraint.
This problem is easily written down in a mixed-integer optimization form. But I am interested in the version of the problem as I wrote. Does this problem have an "easy" solution that is guaranteed optimal? Does this problem have a name?
The problem is NP-hard, by reduction from the partition problem. Given integers $x_1,\dots,x_n$ forming an instance of the partition problem, you can set all $w_{ij}$ to $2/(x_1+\dots+x_n)$, set $|J|=2$, $C_j=1$; then any solution to your problem is a solution to the partition problem. Therefore, there is no "easy" algorithm that is guaranteed to find the optimal solution (unless P=NP).