A hierarchy of semidefinite relaxations for completely positive tensor optimization problems

Zhou, Anwa; Fan, Junxuan

doi:10.1007/s10898-019-00751-8

J Glob Optim

2019

DOI: 10.1007/s10898-019-00751-8

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A hierarchy of semidefinite relaxations for completely positive tensor optimization problems

Anwa Zhou

Junxuan Fan

Abstract: In this paper, we study the completely positive (CP) tensor program, which is a linear optimization problem with the cone of CP tensors and some linear constraints. We reformulate it as a linear program over the cone of moments, then construct a hierarchy of semidefinite relaxations for solving it. We also discuss how to find a best CP approximation of a given tensor. Numerical experiments are presented to show the efficiency of the proposed methods.

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“…The two cones CP n and COP n play an important role in the field of copositive optimization, which establishes a connection between discrete and continuous optimization, and has many real-world applications, see [6,8,12,16]. See [22,23,29] for some recent uses of completely positive tensors in the field of optimization. From [23, Proposition 1], we know that CP n,d is also a proper cone for d ≥ 3.…”

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Lower bounds for maximal cp-ranks of completely positive matrices and tensors

Schachinger

2020

ELA

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Let $p_n$ denote the maximal cp-rank attained by completely positive $n\times n$ matrices. Only lower and upper bounds for $p_n$ are known, when $n\ge6$, but it is known that $p_n=\frac{n^2}2\big(1+o(1)\big)$, and the difference of the current best upper and lower bounds for $p_n$ is of order $\mathcal{O}\big(n^{3/2}\big)$. In this paper, that gap is reduced to $\mathcal{O}\big(n\log\log n\big)$. To achieve this result, a sequence of generalized ranks of a given matrix A has to be introduced. Properties of that sequence and its generating function are investigated. For suitable A, the $d$th term of that sequence is the cp-rank of some completely positive tensor of order $d$. This allows the derivation of asymptotically matching lower and upper bounds for the maximal cp-rank of completely positive tensors of order $d>2$ as well.

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Lower bounds for maximal cp-ranks of completely positive matrices and tensors

Schachinger

2020

ELA

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Approximation hierarchies for copositive cone over symmetric cone and their comparison

Nishijima

Nakata

2023

J Glob Optim

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We first provide an inner-approximation hierarchy described by a sum-of-squares (SOS) constraint for the copositive (COP) cone over a general symmetric cone. The hierarchy is a generalization of that proposed by Parrilo (Structured semidefinite programs and semialgebraic geometry methods in Robustness and optimization, Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 2000) for the usual COP cone (over a nonnegative orthant). We also discuss its dual. Second, we characterize the COP cone over a symmetric cone using the usual COP cone. By replacing the usual COP cone appearing in this characterization with the inner- or outer-approximation hierarchy provided by de Klerk and Pasechnik (SIAM J Optim 12(4):875–892, https://doi.org/10.1137/S1052623401383248, 2002) or Yıldırım (Optim Methods Softw 27(1):155–173, https://doi.org/10.1080/10556788.2010.540014, 2012), we obtain an inner- or outer-approximation hierarchy described by semidefinite but not by SOS constraints for the COP matrix cone over the direct product of a nonnegative orthant and a second-order cone. We then compare them with the existing hierarchies provided by Zuluaga et al. (SIAM J Optim 16(4):1076–1091, https://doi.org/10.1137/03060151X, 2006) and Lasserre (Math Program 144:265–276, https://doi.org/10.1007/s10107-013-0632-5, 2014). Theoretical and numerical examinations imply that we can numerically increase a depth parameter, which determines an approximation accuracy, in the approximation hierarchies derived from de Klerk and Pasechnik (SIAM J Optim 12(4):875–892, https://doi.org/10.1137/S1052623401383248, 2002) and Yıldırım (Optim Methods Softw 27(1):155–173, https://doi.org/10.1080/10556788.2010.540014, 2012), particularly when the nonnegative orthant is small. In such a case, the approximation hierarchy derived from Yıldırım (Optim Methods Softw 27(1):155–173, https://doi.org/10.1080/10556788.2010.540014, 2012) can yield nearly optimal values numerically. Combining the proposed approximation hierarchies with existing ones, we can evaluate the optimal value of COP programming problems more accurately and efficiently.

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A hierarchy of semidefinite relaxations for completely positive tensor optimization problems

Cited by 2 publications

References 35 publications

Lower bounds for maximal cp-ranks of completely positive matrices and tensors

Lower bounds for maximal cp-ranks of completely positive matrices and tensors

Approximation hierarchies for copositive cone over symmetric cone and their comparison

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