Abstract
This paper is concerned with the stochastic structured tensors to stochastic complementarity problems. The definitions and properties of stochastic structured tensors, such as the stochastic strong P-tensors, stochastic P-tensors, stochastic \(P_{0}\)-tensors, stochastic strictly semi-positive tensors and stochastic S-tensors are given. It is shown that the expected residual minimization formulation (ERM) of the stochastic structured tensor complementarity problem has a nonempty and bounded solution set. Interestingly, we partially answer the open questions proposed by Che et al. (Optim Lett 13:261–279, 2019). We also consider the expected value method of stochastic structured tensor complementarity problem with finitely many elements probability space. Finally, based on the expected residual minimization formulation (ERM) of the stochastic structured tensor complementarity problem, a projected gradient method is proposed for solving the stochastic structured tensor complementarity problem and the related numerical results are also given to show the efficiency of the proposed method.
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Shouqiang Du is supported by the National Natural Science Foundation of China under Grant 11671220 and Shandong Provincial Nature Science Foundation under grant ZR2016AM29. Maolin Che is supported by the Fundamental Research Funds for the Central Universities under Grant JBK1801058 and the National Natural Science Foundation of China under Grant 11901471. Yimin Wei is supported by the National Natural Science Foundation of China under Grant 11771099 and Innovation Program of Shanghai Municipal Education Commission.
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Du, S., Che, M. & Wei, Y. Stochastic structured tensors to stochastic complementarity problems. Comput Optim Appl 75, 649–668 (2020). https://doi.org/10.1007/s10589-019-00144-3
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DOI: https://doi.org/10.1007/s10589-019-00144-3
Keywords
- Stochastic tensor complementarity problem
- Stochastic strong P-tensors
- Stochastic P-tensors
- Stochastic \(P_0\)-tensors
- Stochastic strictly semi-positive tensors
- Stochastic S-tensors
- Projected gradient method