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Introduction to Tensor Variational Inequalities

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Yong Wang, Zheng-Hai Huang, Liqun Qi

Optimization and Control · Submitted 26 Jan 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim To this end, we first introduce two classes of structured tensors and discuss some related properties; and then, we show that the TVI has the property of global uniqueness and solvability under some assumptions, which is different from the existed result for the general variational inequality.

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arXiv ID1701.07677 (version 1)
Pith IDKLOGCMRK, Pith's permanent record of this paper and its review
CitationLiqun Qi, Yong Wang, Zheng-Hai Huang. "Introduction to Tensor Variational Inequalities." arXiv:1701.07677 (2017). https://pith.science/paper/KLOGCMRK
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,296 input tokens.

What the abstract itself states, as probabilities: claims first 0.11 · states limitation 0.27 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.85 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

In this paper, we introduce a class of variational inequalities, where the involved function is the sum of an arbitrary given vector and a homogeneous polynomial defined by a tensor; and we call it the tensor variational inequality (TVI). The TVI is a natural extension of the affine variational inequality and the tensor complementarity problem. We show that a class of multi-person noncooperative games can be formulated as a TVI. In particular, we investigate the global uniqueness and solvability of the TVI. To this end, we first introduce two classes of structured tensors and discuss some related properties; and then, we show that the TVI has the property of global uniqueness and solvability under some assumptions, which is different from the existed result for the general variational inequality.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In this paper, we introduce a class of variational inequalities, where the involved function is the sum of an arbitrary given vector and a homogeneous polynomial defined by a tensor; and we call it the tensor variational inequality (TVI). contribution 1.00 0.91 0.08 asserted 0.99
s1 The TVI is a natural extension of the affine variational inequality and the tensor complementarity problem. background 0.71 0.19 0.05 asserted 1.00
s2 We show that a class of multi-person noncooperative games can be formulated as a TVI. contribution 0.85 0.93 0.08 asserted 1.00
s3 In particular, we investigate the global uniqueness and solvability of the TVI. contribution 0.97 0.91 0.19 asserted 0.88
s4 To this end, we first introduce two classes of structured tensors and discuss some related properties; and then, we show that the TVI has the property of global uniqueness and solvability under some assumptions, which is different from the existed result for the general variational inequality. contribution 0.50 0.95 0.68 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.91 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

In this paper, we introduce a class of variational inequalities, where the involved function is the sum of an arbitrary given vector and a homogeneous polynomial defined by a tensor; and we call it the tensor variational inequality (TVI). The TVI is a natural extension of the affine variational inequality and the tensor complementarity problem. We show that a class of multi-person noncooperative games can be formulated as a TVI. In particular, we investigate the global uniqueness and solvability of the TVI. To this end, we first introduce two classes of structured tensors and discuss some related properties; and then, we show that the TVI has the property of global uniqueness and solvability under some assumptions, which is different from the existed result for the general variational inequality.

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What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites

Cites as background 6 papers
  • Recently, a specific subclass of the CPs, called the tensor complementarity problem (TCP) [ref] , has attracted much attention; and many theoretical results about the properties of the solution set of TCP have been developed, including existence of solution [ref] , global uniqueness of solution [ref] , boundedness of solution set [ref] , stability of solution [ref] , sparsity of solution [ref] , and so on.

    Properties of Tensor Complementarity Problem and Some Classes of Structured Tensors · Introduction · 0.92

  • Recently, a specific subclass of the CPs, called the tensor complementarity problem (TCP) [ref] , has attracted much attention; and many theoretical results about the properties of the solution set of TCP have been developed, including existence of solution [ref] , global uniqueness of solution [ref] , boundedness of solution set [ref] , stability of solution [ref] , sparsity of solution [ref] , and so on.

    P-Tensors, P$_0$-Tensors, and Tensor Complementarity Problem · Introduction · 0.92

  • Recently, a specific subclass of the CPs, called the tensor complementarity problem (TCP) [ref] , has attracted much attention; and many theoretical results about the properties of the solution set of TCP have been developed, including existence of solution [ref] , global uniqueness of solution [ref] , boundedness of solution set [ref] , stability of solution [ref] , sparsity of solution [ref] , and so on.

    Eigenvalues and structured properties of P-tensors · Introduction · 0.92

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