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.

中文翻译：

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随机互补张量的随机结构张量

本文涉及随机互补张量的随机结构化张量。随机结构张量，如随机强的定义和属性P -tensors，随机P -tensors，随机\（P_ {0} \） -tensors，随机严格半正张量和随机小号给出张量。结果表明，随机结构化张量互补问题的期望残差最小化公式（ERM）具有非空有界解集。有趣的是，我们部分地回答了Che等人提出的开放性问题。（Optim Lett 13：261–279，2019）。我们还考虑了具有有限多个元素概率空间的随机结构化张量互补问题的期望值方法。最后，基于随机结构化张量互补问题的期望残差最小化公式（ERM），提出了一种投影梯度法来求解随机结构化张量互补问题，并给出了相关数值结果，表明了该方法的有效性。 。