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Stochastic \(R_0\) tensors to stochastic tensor complementarity problems

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Abstract

The main purpose of this paper is devoted to an introduction of the stochastic tensor complementarity problem. We consider the expected residual minimization formulation of the stochastic tensor complementarity problem. We show that the solution set of the expected residual minimization problem is nonempty and bounded, if the associated tensor is an \(R_0\) tensor. We also prove that the associated tensor being a stochastic \(R_0\) tensor is a necessary and sufficient condition for the existence of the solution of the expected residual minimization problem to be nonempty and bounded.

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Notes

  1. For a given \({\mathbf {x}}\in {\mathbb {R}}^I\), \({\mathrm{diag}}({\mathbf {x}})\in {\mathbb {R}}^{I\times I}\) is a diagonal matrix such that its diagonal entries equal to the entries of \({\mathbf {x}}\).

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Acknowledgements

The authors would like to thank the Editor-in-Chief Prof. Oleg Prokopyev, and two referees for their detailed comments and suggestions which greatly improve the presentation of our paper.

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Authors and Affiliations

  1. School of Economic Mathematics, Southwest University of Finance and Economics, Chengdu, 611130, People’s Republic of China

    Maolin Che

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong

    Liqun Qi

  3. School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, People’s Republic of China

    Yimin Wei

Authors
  1. Maolin Che
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  2. Liqun Qi
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  3. Yimin Wei
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Corresponding author

Correspondence to Yimin Wei.

Additional information

Liqun Qi is supported by the Hong Kong Research Grant Council (Grant Nos. PolyU 501913, 15302114, 15300715 and 15301716). Yimin Wei is supported by the National Natural Science Foundation of China under Grant 11771099 and Shanghai Municipal Education Commission.

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Che, M., Qi, L. & Wei, Y. Stochastic \(R_0\) tensors to stochastic tensor complementarity problems. Optim Lett 13, 261–279 (2019). https://doi.org/10.1007/s11590-018-1362-7

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  • DOI: https://doi.org/10.1007/s11590-018-1362-7

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