An Explicit SOS Decomposition of A Fourth Order Four Dimensional Hankel Tensor with A Symmetric Generating Vector

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1 arxiv: v1 [math.oc] 8 Mar 2015 An Explicit SOS Decomposition of A Fourth Order Four Dimensional Hankel Tensor with A Symmetric Generating Vector Yannan Chen Liqun Qi Qun Wang September 25, 2018 Abstract In this note, we construct explicit SOS decomposition of A Fourth Order Four Dimensional Hankel Tensor with A Symmetric Generating Vector, at the critical value. This is a supplementary note to Paper [3]. Key words: Hankel tensors, generating vectors, sum of squares, positive semidefiniteness. AMS subject classifications (2010): 15A18; 15A69 We construct an explicit SOS decomposition for the homogeneous polynomial f(x). Let u := ( x 2 1,x2 2,x2 3,x2 4,x ) 1x 3,x 2 x 4,x 1 x 2,x 3 x 4,x 2 x 3,x 1 x 4, be a basis of quadratics. Then, f(x) is SOS if and only if there exists a PSD matrix C such that f(x) = u Cu. Inspired by the Cholecky decomposition of C, we define the parameterized SOS decomposition of f(x) as follows f(x) = 10 q 2 k (x), School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, China. ynchen@zzu.edu.cn (Y. Chen) This author s work was supported by the National Natural Science Foundation of China (Grant No ) and the Development Foundation for Excellent Youth Scholars of Zhengzhou University (Grant No ). Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. maqilq@polyu.edu.hk (L. Qi). This author s work was partially supported by the Hong Kong Research Grant Council (Grant No. PolyU , , and ). Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. wangqun876@gmail.com (Q. Wang). 1

2 where q 1 (x) = α 11 x 2 1 +α 12 x 2 2 +α 13 x 2 3 +α 14 x 2 4 +α 15 x 1 x 3 +α 16 x 2 x 4 +α 17 x 1 x 2 +α 18 x 3 x 4 +α 19 x 2 x 3 +α 1,10 x 1 x 4, q 2 (x) = α 22 x 2 2 +α 23x 2 3 +α 24x 2 4 +α 25x 1 x 3 +α 26 x 2 x 4 +α 27 x 1 x 2 +α 28 x 3 x 4 +α 29 x 2 x 3 +α 2,10 x 1 x 4, q 3 (x) = α 33 x 2 3 +α 34x 2 4 +α 35x 1 x 3 +α 36 x 2 x 4 +α 37 x 1 x 2 +α 38 x 3 x 4 +α 39 x 2 x 3 +α 3,10 x 1 x 4, q 4 (x) = α 44 x 2 4 +α 45x 1 x 3 +α 46 x 2 x 4 +α 47 x 1 x 2 +α 48 x 3 x 4 +α 49 x 2 x 3 +α 4,10 x 1 x 4, q 5 (x) = α 55 x 1 x 3 +α 56 x 2 x 4 +α 57 x 1 x 2 +α 58 x 3 x 4 +α 59 x 2 x 3 +α 5,10 x 1 x 4, q 6 (x) = α 66 x 2 x 4 +α 67 x 1 x 2 +α 68 x 3 x 4 +α 69 x 2 x 3 +α 6,10 x 1 x 4, q 7 (x) = α 77 x 1 x 2 +α 78 x 3 x 4 +α 79 x 2 x 3 +α 7,10 x 1 x 4, q 8 (x) = α 88 x 3 x 4 +α 89 x 2 x 3 +α 8,10 x 1 x 4, q 9 (x) = α 99 x 2 x 3 +α 9,10 x 1 x 4, q 10 (x) = α 10,10 x 1 x 4. 2

3 The involved 55 parameters α i,j,(i j), satisfies the following 35 equality constraints v 0 = α11, 2 (1) 4v 1 = 2α 11 α 17, (2) 4v 2 = 2α 11 α 15, (3) 7 6v 2 = 2α 11 α 12 + αk7, 2 (4) 4v 3 = 2α 11 α 1,10, (5) 12v 3 = 2α 11 α α k5 α k7, (6) 4v 3 = 2 α k2 α k7, (7) 12 = 2α 11 α = 2α 11 α = 2 1 = 7 α k7 α k10, (8) αk5 2, (9) α k2 α k α k7 α k9, (10) αk2 2, (11) α 11 α v 5 = 2 α k5 α k,10, (12) α k2 α k10 +2 α k3 α k7 +2 α k6 α k7, (13) α k5 α k9, (14) α k2 α k9, (15) 6v 6 = 2α 11 α v 6 = 2 4v 6 = 2 10 α k5 α k6 +2 α 2 k10, (16) 7 9 α k7 α k8 +2 α k9 α k10, (17) α k3 α k5, (18) 3

4 4v 6 = 2 6v 6 = 2 4v 5 = 2 12 = 2 6 = 2 12 = 2 1 = 4v 3 = 2 12v 3 = 2 4v 3 = 2 4v 2 = 2 6v 2 = 2 4v 1 = 2 v 0 = α k2 α k6, (19) α k2 α k3 + α k4 α k αk9 2, (20) α k3 α k10 +2 α k2 α k8 +2 α k6 α k10, (21) α k5 α k8, (22) α k6 α k9, (23) α k3 α k9, (24) α k4 α k5 +2 α k2 α k4 +2 α k3 α k α k8 α k10, (25) αk6 2, (26) 8 α k8 α k9, (27) αk3 2, (28) α k4 α k10, (29) α k4 α k9 +2 α k6 α k8, (30) α k3 α k8, (31) α k4 α k6, (32) α k3 α k αk8 2, (33) α k4 α k8, (34) α 2 k4. (35) 4

5 Giving fixed v 2,v 6,v 1,v 3,v 5, we denote M 1 (v 2,v 6,v 1,v 3,v 5 ) as the minimum of the following optimization problems: M 1 (v 2,v 6,v 1,v 3,v 5 ) := min v 0 s.t. equality constraints (1)-(35). (36) By simple algebraic derivation, we have the following theorem. Theorem 1 Supposethatthe assumptions(2) and(3)in [3]hold. Thenif v 0 M 1 (v 2,v 6,v 1,v 3,v 5 ), A is SOS and 10 f(x) = q k (x) 2. Thus, M 1 (v 2,v 6,v 1,v 3,v 5 ) M 0 (v 2,v 6,v 1,v 3,v 5 ). References [1] A.A. Ahmadi and P.A. Parrilo, A convex polynomial that is not sos-convex, Mathematical Programming 135 (2012) [2] R. Badeau and R. Boyer, Fast multilinear singular value decomposition for structured tensors, SIAM Journal on Matrix Analysis and Applications 30 (2008) [3] Y. Chen, L. Qi and Q. Wang, Positive semi-definiteness and sum-of-squares property of fourth order four dimensional Hankel tensors, arxiv: , March [4] G. Chesi, On the gap between positive polynomials and SOS of polynomials, IEEE Transactions on Automatic Control 52 (2007) [5] M.D. Choi and T.Y. Lam, Extremal positive semidefinite forms, Mathematische Annalen 231 (1977) [6] W. Ding, L. Qi and Y. Wei, Fast Hankel tensor-vector products and application to exponential data fitting, Numerical Linear Algebra with Applications, (2015), DOI: /nla [7] C. Fidalgo and A. Kovacec, Positive semidefinite diagonal minus tail forms are sums of squares, Mathematische Zeitschrift 269 (2011) [8] D. Hilbert, Über die Darstellung definiter Formen als Summe von Formenquadraten, Mathematical Annals, 32 (1888)

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