Given a data matrix $D\in\mathbb{R}^{N \times N}$
Can one construct another matrix $M$ that for all permutation matrices $Q^A$,$Q^B$,
if $[\sum_i\sum_j (Q^A_{ij}D_{ij})]^2 \geq [\sum_i\sum_j (Q^B_{ij}D_{ij})]^2$ then $\sum_i\sum_j (Q^A_{ij}M_{ij}) \geq \sum_i\sum_j (Q^B_{ij}M_{ij})$?
p.s. Sorry I don't know how to name the problem so the title is just "Question about inearization". If you have any good idea please tell me.