A semidefinite relaxation method for second-order cone tensor eigenvalue complementarity problems
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A semidefinite relaxation method for second-order cone tensor eigenvalue complementarity problems Lulu Cheng1 · Xinzhen Zhang1
· Guyan Ni2
Received: 10 October 2019 / Accepted: 27 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper discusses second-order cone tensor eigenvalue complementarity problem. We reformulate second-order cone tensor eigenvalue complementarity problem as two constrained polynomial optimizations. For these two reformulated optimizations, Lasserre-type semidefinite relaxation methods are proposed to compute all second-order cone tensor complementarity eigenpairs. The proposed algorithms terminate when there are finitely many second-order cone complementarity eigenvalues. Numerical examples are reported to show the efficiency of the proposed algorithms. Keywords Second-order cone · Tensor eigenvalue complementarity · Semidefinite relaxation Mathematics Subject Classification 15A18 · 15A69 · 90C22 · 90C33
1 Introduction For positive integers m and n 1 , n 2 , . . . , n m , an mth order and (n 1 , n 2 , . . . , n m )-dimensional real tensor can be viewed as an array in the space Rn 1 ×n 2 ×···×n m . Such a tensor A can be indexed as A = (Ai1 ...im ), 1 ≤ i j ≤ n j ,
j = 1, . . . , m.
When n 1 = · · · = n m = n, A is called an mth order n-dimensional square tensor. In such case, the tensor space Rn 1 ×···×n m is denoted as Tm (Rn ). A tensor in Tm (Rn ) is said to be symmetric if its entries are invariant under permutations of indices. That is, Ai1 ...im = A j1 ... jm for any
B
Xinzhen Zhang [email protected] Lulu Cheng [email protected] Guyan Ni [email protected]
1
School of Mathematics, Tianjin University, Tianjin 300354, China
2
Department of Mathematics, National University of Defense Technology, Changsha, Hunan, China
123
Journal of Global Optimization
permutation ( j1 , . . . , jm ) of index (i 1 , . . . , i m ). For A ∈ Tm (Rn ) and x := (x1 , . . . , xn )T ∈ Rn , we use the notation ⎧ m := Ai1 i2 ...im xi1 xi2 . . . xim , ⎪ ⎨ Ax 1≤i m ≤n 1 ,...,i m−1 := ⎪ A ji2 ...im xi2 . . . xim . ⎩ Ax 1≤i 2 ,...,i m ≤n
j=1,...,n
Clearly, Ax m−1 ∈ Rn is an n-dimensional vector. In this paper, we consider the second-order cone tensor eigenvalue complementarity problem (SOCTEiCP): for two given tensors A, B ∈ Tm (Rn+1 ), SOCTEiCP is to find a nonzero vector x with a real number λ such that K x ⊥ (Ax m−1 − λB x m−1 ) ∈ K∗ ,
(1.1)
where K is the second-order cone (or Lorentz cone, ice-cream cone in the literature) defined by K := {x = (z, t) ∈ Rn+1 : z2 ≤ t}, and here K∗ is the dual cone of K, defined by K∗ := {y| x T y ≥ 0, ∀x ∈ K}. For (λ, x) satisfying (1.1), λ is called an SOC-complementarity eigenvalue and x is called an SOC-complementarity eigenvector of (A, B), respectively. Furthermore, we call such (λ, x) an SOC-complementarity eigenpair of (A, B). For a general closed and convex cone, problem (1.1) is called the tensor generalized eigenvalue complementarity problem (TGEiCP), introduced and studied in [20]. When th
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