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May 7, 2019 at 3:52 comment added ryanseadub To add to the completely positive part of the answer: while checking membership of the completely positive cone is NP-hard, you can use the set of doubly non-negative matrices (i.e. positive semidefinite and entry-wise non-negative matrices): this is a valid outer approximation which is semidefinitely representable, and agrees with the copositive cone for $n \leq 4$, see sciencedirect.com/science/article/pii/S002437950900281X.
May 6, 2019 at 9:41 vote accept πr8
May 6, 2019 at 9:41 comment added πr8 Perfect - thank you!
May 6, 2019 at 9:39 history answered Nathaniel Johnston CC BY-SA 4.0