Luo Meiju, Zhang Yan
School of Mathematics, Liaoning University, Liaoning, China.
J Inequal Appl. 2018;2018(1):77. doi: 10.1186/s13660-018-1674-2. Epub 2018 Apr 10.
In this paper, we consider stochastic second-order-cone complementarity problems (SSOCCP). We first use the so-called second-order-cone complementarity function to present an expected residual minimization (ERM) model for giving reasonable solutions of SSOCCP. Then, we introduce a smoothing function, by which we obtain a smoothing approximate ERM model. We further show that the global solution sequence and weak stationary point sequence of this smoothing approximate ERM model converge to the global solution and the weak stationary point of the original ERM model as the smoothing parameter tends to zero respectively. Moreover, since the ERM formulation contains an expectation, we employ a sample average approximate method for solving the smoothing ERM model. As the convergence analysis, we first show that the global optimal solution of this smoothing sample average approximate problem converges to the global optimal solution of the ERM problem with probability one. Subsequently, we consider the weak stationary points' convergence results of this smoothing sample average approximate problem of ERM model. Finally, some numerical examples are given to explain that the proposed methods are feasible.
在本文中，我们考虑随机二阶锥互补问题（SSOCCP）。我们首先使用所谓的二阶锥互补函数来提出一个期望残差最小化（ERM）模型，以给出SSOCCP的合理解。然后，我们引入一个平滑函数，通过它得到一个平滑近似ERM模型。我们进一步证明，当平滑参数趋于零时，这个平滑近似ERM模型的全局解序列和弱驻点序列分别收敛到原始ERM模型的全局解和弱驻点。此外，由于ERM公式包含一个期望，我们采用样本平均近似方法来求解平滑ERM模型。作为收敛性分析，我们首先表明这个平滑样本平均近似问题的全局最优解以概率1收敛到ERM问题的全局最优解。随后，我们考虑ERM模型的这个平滑样本平均近似问题的弱驻点收敛结果。最后，给出一些数值例子来说明所提出的方法是可行的。
