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Column sufficient tensors and tensor complementarity problems
Haibin CHEN, Liqun QI, Yisheng SONG
Front. Math. China. 2018, 13 (2): 255-276.
https://doi.org/10.1007/s11464-018-0681-4
Stimulated by the study of sufficient matrices in linear complementarity problems, we study column sufficient tensors and tensor complementarity problems. Column sufficient tensors constitute a wide range of tensors that include positive semi-definite tensors as special cases. The inheritance property and invariant property of column sufficient tensors are presented. Then, various spectral properties of symmetric column sufficient tensors are given. It is proved that all H-eigenvalues of an even-order symmetric column sufficient tensor are nonnegative, and all its Z-eigenvalues are nonnegative even in the odd order case. After that, a new subclass of column sufficient tensors and the handicap of tensors are defined. We prove that a tensor belongs to the subclass if and only if its handicap is a finite number. Moreover, several optimization models that are equivalent with the handicap of tensors are presented. Finally, as an application of column sufficient tensors, several results on tensor complementarity problems are established.
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Castelnuovo-Mumford regularity and projective dimension of a squarefree monomial ideal
Lizhong CHU, Shisen LIU, Zhongming TANG
Front. Math. China. 2018, 13 (2): 277-286.
https://doi.org/10.1007/s11464-017-0680-x
Let S = K[x1, x2, . . . , xn] be the polynomial ring in n variables over a field K, and let I be a squarefree monomial ideal minimally generated by the monomials u1, u2, . . . , um. Let w be the smallest number t with the property that for all integers such that lcm lcm(u1, u2, . . . , um). We give an upper bound for Castelnuovo-Mumford regularity of I by the bigsize of I. As a corollary, the projective dimension of I is bounded by the number w.
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Error bounds of Lanczos approach for trust-region subproblem
Leihong ZHANG, Weihong YANG, Chungen SHEN, Jiang FENG
Front. Math. China. 2018, 13 (2): 459-481.
https://doi.org/10.1007/s11464-018-0687-y
Because of its vital role of the trust-region subproblem (TRS) in various applications, for example, in optimization and in ill-posed problems, there are several factorization-free algorithms for solving the large-scale sparse TRS. The truncated Lanczos approach proposed by N. I. M. Gould, S. Lucidi, M. Roma, and P. L. Toint [SIAM J. Optim., 1999, 9: 504–525] is a natural extension of the classical Lanczos method for the symmetric linear system and eigenvalue problem and, indeed follows the classical Rayleigh-Ritz procedure for eigenvalue computations. It consists of 1) projecting the original TRS to the Krylov subspaces to yield smaller size TRS’s and then 2) solving the resulted TRS’s to get the approximates of the original TRS. This paper presents a posterior error bounds for both the global optimal value and the optimal solution between the original TRS and their projected counterparts. Our error bounds mainly rely on the factors from the Lanczos process as well as the data of the original TRS and, could be helpful in designing certain stopping criteria for the truncated Lanczos approach.
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13 articles
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