An algorithm for decomposing a non-negative polynomial as a sum of squares of rational functions

Le, Thanh Hieu; Barel, Marc Van

doi:10.1007/s11075-014-9903-3

Cited by 4 publications

(1 citation statement)

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“…On the other hand, an interesting feature of SOS tensor decomposition is checking whether a given even order symmetric tensor has SOS decomposition or not can be verified by solving a semi-definite programming problem (see for example [14]), and hence, can be validated efficiently. SOS tensor decomposition has a close connection with SOS polynomials, and SOS polynomials are very important in polynomial theory [3,4,12,13,36,41] and polynomial optimization [16,21,22,23,35,42]. It is known that an even order symmetric tensor having SOS decomposition is positive semi-definite, but the converse is not true in general.…”

Section: Introductionmentioning

confidence: 99%

SOS tensor decomposition: Theory and applications

Chen

2016

Communications in Mathematical Sciences

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In this paper, we examine structured tensors which have sum-of-squares (SOS) tensor decomposition, and study the SOS-rank of SOS tensor decomposition. We first show that several classes of even order symmetric structured tensors available in the literature have SOS tensor decomposition. These include positive Cauchy tensors, weakly diagonally dominated tensors, B0-tensors, double B-tensors, quasi-double B0-tensors, M B0-tensors, H-tensors, absolute tensors of positive semi-definite Z-tensors and extended Z-tensors. We also examine the SOS-rank of SOS tensor decomposition and the SOS-width for SOS tensor cones. The SOS-rank provides the minimal number of squares in the SOS tensor decomposition, and, for a given SOS tensor cone, its SOS-width is the maximum possible SOS-rank for all the tensors in this cone. We first deduce an upper bound for general tensors that have SOS decomposition and the SOS-width for general SOS tensor cone using the known results in the literature of polynomial theory. Then, we provide an explicit sharper estimate for the SOS-rank of SOS tensor decomposition with bounded exponent and identify the SOS-width for the tensor cone consisting of all tensors with bounded exponent that have SOS decompositions. Finally, as applications, we show how the SOS tensor decomposition can be used to compute the minimum H-eigenvalue of an even order symmetric extended Z-tensor and test the positive definiteness of an associated multivariate form. Numerical experiments are also provided to show the efficiency of the proposed numerical methods ranging from small size to large size numerical examples.

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