Stochastic \(R_0\) matrix, which is a generalization of \(R_0\) matrix, is instrumental in the study of the stochastic linear complementarity problem (SLCP). In this paper, we focus on the expected residual minimization (ERM) of the SLCP via the Fischer–Burmeister (FB) function, which enjoys better properties of the continuity and differentiability than the min-function-based ERM when the involved matrix is a stochastic \(R_0\) matrix. First, we prove that the solution set of the FB-function-based ERM is nonempty and bounded if and only if the involved matrix is a stochastic \(R_0\) matrix. Second, we show that its objective function is continuously differentiable in \({\mathbb {R}}^n\), which makes it possible for us to design the gradient-type algorithm. Finally, we implement the numerical experiments generated randomly via sample average approximation, and the numerical results indicate that the optimal solutions of the FB-function-based ERM are better than that of the min-function-based ERM in terms of preserving the nonnegativity of the involved linear function, especially when the involved matrix is a constant matrix.

随机\（R_0 \）矩阵，这是一个一般化\（R_0 \）矩阵，是随机线性互补问题（SLCP）的研究工具。在本文中，我们通过 Fischer-Burmeister (FB) 函数关注 SLCP 的预期残差最小化 (ERM)，当所涉及的矩阵为随机\(R_0\)矩阵。首先，我们证明了基于 FB 函数的 ERM 的解集是非空且有界的，当且仅当所涉及的矩阵是随机\(R_0\)矩阵。其次，我们证明它的目标函数在\({\mathbb {R}}^n\) 中是连续可微的，这使我们可以设计梯度型算法。最后，我们实现了通过样本平均逼近随机生成的数值实验，数值结果表明基于 FB 函数的 ERM 的最优解在保留非负性方面优于基于最小函数的 ERM 的最优解。涉及的线性函数，特别是当涉及的矩阵是常数矩阵时。