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Mixed polynomial variational inequalities

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Abstract

The goal of this paper is to introduce a class of mixed polynomial variational inequalities, which is a natural generalization of the affine variational inequality and the tensor variational inequality, and a special case of the mixed variational inequality. It is shown that a class of polynomial optimization problem and a class of m-person noncooperative game can be reformulated as a mixed polynomial variational inequality. Firstly, some classes of structured tensor tuples are introduced and the relationship between them is discussed. Then, a new asymptotic function (denoted by m-asymptotic function) is introduced and some basic properties are investigated. An equivalent characterization for the nonexistence of solutions is given by using the exceptional family of elements. Finally, the nonemptiness and compactness of the solution sets of the mixed polynomial variational inequalities with some special structured tensors and m-asymptotic function are proved and then the uniqueness of the solution is further investigated.

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Acknowledgements

The authors are grateful to the Editor and the reviewers for their helpful comments and suggestions, which have improved the presentation of the paper. This work was partially supported by National Natural Science Foundation of China (No.11961006) and Guangxi Natural Science Foundation (2020GXNSFAA159100).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, People’s Republic of China

    Tong-tong Shang

  2. School of Mathematics and Physics, Guangxi Minzu University, Nanning, 530006, Guangxi, People’s Republic of China

    Tong-tong Shang

  3. School of Mathematics and Physics, Center for Applied Mathematics of Guangxi, Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi Minzu University, Guangxi, 530006, People’s Republic of China

    Guo-ji Tang

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  1. Tong-tong Shang
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Shang, Tt., Tang, Gj. Mixed polynomial variational inequalities. J Glob Optim 86, 953–988 (2023). https://doi.org/10.1007/s10898-023-01298-5

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