I have the following inequality that I would want to linearize. Consider that $r_{ij}, x_{ij}, y_{ij}$ are binary variables defined for every pair of nodes $(i,j) \in A$. Also, I have a set of nodes $V$ which its used later.
The following constraint holds: $$r_{ij} \leq \sum_{k \ne i, k \ne j}x_{ik}y_{jk}\qquad \forall (i,j) \in A$$
At first I thought that I should introduce a new family of 3-index binary variables $l^k_{ij}$ and define: $$l^k_{ij} \leq x_{ik}\qquad \forall (i,j) \in A, k\in V: i \ne k, j\ne k\\l^k_{ij}\leq y_{jk}\qquad \forall (i,j) \in A,k \in V:i\ne k, j\ne k\\ l^k_{ij} \geq x_{ik}+y_{jk}-1 \qquad \forall (i,j) \in A,k \in V:i\ne k, j\ne k$$
My ultimate goal is to be able to come up with a linearization that maintains the original number of indexes (although so far i haven't been able to do so and I think it may not be possible).
I've read this survey Transformation and Linearization Techniques in Optimization: A State-of-the-Art Survey and was wondering if there are any others papers like it that I could check as inspiration.