Complexity of the Positive Semidefinite Matrix Completion Problem with a Rank Constraint [chapter]

Marianna E.-Nagy, Monique Laurent, Antonios Varvitsiotis
2013 Fields Institute Communications  
We consider the decision problem asking whether a partial rational symmetric matrix with an all-ones diagonal can be completed to a full positive semidefinite matrix of rank at most k. We show that this problem is N P -hard for any fixed integer k ≥ 2. Equivalently, for k ≥ 2, it is N P -hard to test membership in the rank constrained elliptope E k (G), i.e., the set of all partial matrices with off-diagonal entries specified at the edges of G, that can be completed to a positive semidefinite
more » ... trix of rank at most k. Additionally, we show that deciding membership in the convex hull of E k (G) is also N P -hard for any fixed integer k ≥ 2. * M.E.Nagy@cwi.nl † M.Laurent@cwi.nl ‡ A.Varvitsiotis@cwi.nl arXiv:1203.6602v2 [math.OC]
doi:10.1007/978-3-319-00200-2_7 fatcat:5mpvfe5zkfhahjyfii7ut3jw2a