Positive semi-definiteness and sum-of-squares property of fourth order four dimensional Hankel tensors

摘要

A symmetric positive semi-definite (PSD) tensor, which is not sum-of-squares (SOS), is called a PSD non-SOS (PNS) tensor. Is there a fourth order four dimensional PNS Hankel tensor? The answer for this question has both theoretical and practical significance. Under the assumptions that the generating vector v of a Hankel tensor A is symmetric and the fifth element v(4) of v is fixed at 1, we show that there are two surfaces M-0 and N-0 with the elements v(2), v(6), v(1), v(3), v(5) of v as variables, such that M-0 >= N-0, A is SOS if and only if v(0) >= M-0, and A is PSD if and only if v(0) >= N-0, where v(0) is the first element of v. If M-0 = N-0 for a point P = (v(2), v(6), v(1), v(3), v(5))(T), there are no fourth order four dimensional PNS Hankel tensors with symmetric generating vectors for such v(2), v(6), v(1), v(3), v(5). Then, we call such P a PNS-free point. We prove that a 45-degree planar closed convex cone, a segment, a ray and an additional point are PNS-free. Numerical tests check various grid points and report that they are all PNS-free. (C) 2016 Elsevier B.V. All rights reserved.