Global Uniqueness and Solvability of Tensor Variational Inequalities



In this paper, we consider 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; we call it the tensor variational inequality. The tensor variational inequality 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 tensor variational inequality. In particular, we investigate the global uniqueness and solvability of the tensor variational inequality. To this end, we first introduce two classes of structured tensors and discuss some related properties, and then, we show that the tensor variational inequality has the property of global uniqueness and solvability under some assumptions, which is different from the existing result for the general variational inequality.


Tensor variational inequality Global uniqueness and solvability Noncooperative game Strictly positive definite tensor Exceptionally family of elements 

Mathematics Subject Classification

90C33 90C30 65H10 



The first author’s work is partially supported by the National Natural Science Foundation of China (Grant No. 71572125), the second author’s work is partially supported by the National Natural Science Foundation of China (Grant No. 11431002), and the third author’s work is partially supported by the Hong Kong Research Grant Council (Grant Nos. PolyU 502111, 501212, 501913, and 15302114).


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Authors and Affiliations

  1. 1.School of MathematicsTianjin UniversityTianjinPeople’s Republic of China
  2. 2.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHung Hom, KowloonHong Kong

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