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Existence results for solutions of mixed tensor variational inequalities

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Abstract

By employing the notion of exceptional family of elements, we establish existence results for the mixed tensor variational inequalities. We show that the nonexistence of an exceptional family of elements is a sufficient condition for the solvability of mixed tensor variational inequality. For positive semidefinite mixed tensor variational inequalities, the nonexistence of an exceptional family of elements is proved to be an equivalent characterization of the nonemptiness of the solution sets. We derive several sufficient conditions of the nonemptiness and compactness of the solution sets for the mixed tensor variational inequalities with some special structured tensors. Finally, we show that the mixed tensor variational inequalities can be defined as a class of convex optimization problems.

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Acknowledgements

The authors would like to thank the editor and the anonymous referees for their valuable comments and suggestions which help us to improve this manuscript.

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Correspondence to Jianghua Fan.

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This work was supported by the National Nature Science Foundation of China (71561004) and the Innovation Project of Guangxi Graduate Education of China (XYCSZ2020062)

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Mu, W., Fan, J. Existence results for solutions of mixed tensor variational inequalities. J Glob Optim 82, 389–412 (2022). https://doi.org/10.1007/s10898-021-01080-5

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