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[I'm leaving out constraint 0] I see two levels of decisions in your problem: Group servers into clusters Assign each user to one of these clusters (BTW, this way of seeing the problem is heavily inspired by Facility location problems). The second step is actually the easiest: once the clusters are known, you can just compute each user's gain w.r.t to each ...


They are equivalent except when $x_{i,g}=x_{j,g}=0$, in which case the second linearization incorrectly contributes $-d_{ij}$ to the objective. Assuming $d_{ij} \ge 0$, I recommend a third linearization (relaxing $z$ and omitting two constraints from linearization 1): \begin{align} z_{ijg}&\ge x_{ig}+x_{jg}-1 \\ z_{ijg}&\ge 0 \end{align}


Rank-one constraints are unfortunately not mixed-integer convex representable, as shown in this paper:, although they are quadratically-constrained quadratic representable. If the problem size is not too large, you can try solving it using Gurobi, either directly (for n<=10) or via branch-and-cut (for say n<=50; see ...

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