Strictly semi-positive tensors and the boundedness of tensor complementarity problems

Song, Yisheng; Qi, Liqun

doi:10.1007/s11590-016-1104-7

Cited by 52 publications

(21 citation statements)

References 41 publications

(54 reference statements)

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“…So, we may probe into checking copositivity of tensors and its applications by means of studying this class of semipositive tensors. For its more properties and applications in TCP, see [5,6,9,10,14,15,17,19,30,[46][47][48][51][52][53] and references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Analytical expressions of copositivity for fourth-order symmetric tensors

Song

2020

Anal. Appl.

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In particle physics, scalar potentials have to be bounded from below in order for the physics to make sense. The precise expressions of checking lower bound of scalar potentials are essential, which is an analytical expression of checking copositivity and positive definiteness of tensors given by such scalar potentials. Because the tensors given by general scalar potential are 4th order and symmetric, our work mainly focuses on finding precise expressions to test copositivity and positive definiteness of 4th order tensors in this paper. First of all, an analytically sufficient and necessary condition of positive definiteness is provided for 4th order 2 dimensional symmetric tensors. For 4th order 3 dimensional symmetric tensors, we give two analytically sufficient conditions of (strictly) cpositivity by using proof technique of reducing orders or dimensions of such a tensor. Furthermore, an analytically sufficient and necessary condition of copositivity is 1 showed for 4th order 2 dimensional symmetric tensors. We also give several distinctly analytically sufficient conditions of (strict) copositivity for 4th order 2 dimensional symmetric tensors. Finally, we apply these results to check lower bound of scalar potentials, and to present analytical vacuum stability conditions for potentials of two real scalar fields and the Higgs boson.

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Section: Introductionmentioning

confidence: 99%

Analytical expressions of copositivity for fourth-order symmetric tensors

Song

2020

Anal. Appl.

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“…As a natural extension of the linear complementarity problem, the TCP( , A) emerged from the tensor community in 2015 [1]. In recent several years, the TCP( , A) has been a hot topic and many theoretical results have been obtained, including the nonemptiness and/or compactness of solution set [4][5][6][7][8][9][10][11][12][13][14][15][16], the existence of unique solution [7,8,14,[17][18][19][20][21][22], error bound theory [23][24][25], strict feasibility [22,26], and so on. Several algorithms for solving the TCP( , A) have also been proposed [27][28][29][30][31].…”

Section: Introductionmentioning

confidence: 99%

Solvability of Two Classes of Tensor Complementarity Problems

Yang

2019

Mathematical Problems in Engineering

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In this paper, we first introduce a class of tensors, called positive semidefinite plus tensors on a closed cone, and discuss its simple properties; and then, we focus on investigating properties of solution sets of two classes of tensor complementarity problems. We study the solvability of a generalized tensor complementarity problem with aD-strictly copositive tensor and a positive semidefinite plus tensor on a closed cone and show that the solution set of such a complementarity problem is bounded. Moreover, we prove that a related conic tensor complementarity problem is solvable if the involved tensor is positive semidefinite on a closed convex cone and is uniquely solvable if the involved tensor is strictly positive semidefinite on a closed convex cone. As an application, we also investigate a static traffic equilibrium problem which is reformulated as a concerned complementarity problem. A specific example is also given.

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“…It is easy to see that strong strictly semi-positiveness implies strictly semipositiveness but the converse is not true ( [16]). Song, Qi, and Yu [22,23,25] proved the existence and boundedness of solutions of TCPs for strictly semipositive tensors. Liu, Li, and Vong [16] proved that a TCP possesses the global uniqueness and solvability property if the tensor is strong strictly semipositive, extending a similar result of Bai, Huang, and Wang [3] regarding strong P tensors.…”

Section: Introductionmentioning

confidence: 99%

A Continuation Method for Tensor Complementarity Problems

Han

2018

J Optim Theory Appl

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We introduce a Kojima-Megiddo-Mizuno type continuation method for solving tensor complementarity problems. We show that there exists a bounded continuation trajectory when the tensor is strictly semipositive and any limit point tracing the trajectory gives a solution of the tensor complementarity problem. Moreover, when the tensor is strong strictly semi-positive, tracing the trajectory will converge to the unique solution. Some numerical results are given to illustrate the effectiveness of the method.

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Strictly semi-positive tensors and the boundedness of tensor complementarity problems

Cited by 52 publications

References 41 publications

Analytical expressions of copositivity for fourth-order symmetric tensors

Analytical expressions of copositivity for fourth-order symmetric tensors

Solvability of Two Classes of Tensor Complementarity Problems

A Continuation Method for Tensor Complementarity Problems

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