随机变分不等式的二阶微分方程方法

庄慧婷; 王莉; 孙菊贺; 贾丹娜; 袁艳红

doi:10.3969/j.issn.2095-1248.2023.04.012

沈阳航空航天大学学报 ›› 2023, Vol. 40 ›› Issue (4) : 88 -96. DOI: 10.3969/j.issn.2095-1248.2023.04.012

基础科学和工程

随机变分不等式的二阶微分方程方法

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Second-order differential equation method for solving stochastic variational inequality

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摘要

运用具有正黏性阻尼系数和时间尺度系数的二阶微分方程系统来求解随机变分不等式问题（stochastic variational inequality problem，SVIP）。首先，应用互补函数和样本均值近似（sample average approximation， SAA）方法对原始问题进行等价转换，即将随机变分不等式问题转化为一个方程组，在此基础上建立具有正黏性阻尼系数 $γ t$ 和时间尺度系数 $β t$ 的二阶微分方程系统；其次，研究了该二阶微分方程系统轨迹的收敛性和收敛速率；最后，给出两个数值实验说明该二阶微分方程系统求解随机变分不等式问题的有效性。

Abstract

A system of second-order differential equation with positive viscous damping coefficients and time-scale coefficients was applied to solve the stochastic variational inequality problem （SVIP）. Firstly， the complementary function and the sample average approximation （SAA） method were applied to equate the original problem， and the stochastic variational inequality problem was transformed into a system of equations. Based on this， a second-order differential equation system with positive viscous damping coefficients $γ t$ and time-scale coefficients $β t$ was established. Secondly， the convergence and convergence rate of the trajectory of the second-order differential equation system were obtained. Finally， two numerical experiments were presented to demonstrate the effectiveness of the second-order differential equation system in solving stochastic variational inequality problems.

关键词

随机变分不等式 / 二阶微分方程 / 互补函数 / 样本均值近似方法 / 凸优化问题

Key words

stochastic variational inequality / second-order differential equation / complementary function / sample average approximation method / convex optimization problem

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庄慧婷, 王莉, 孙菊贺, 贾丹娜, 袁艳红. 随机变分不等式的二阶微分方程方法[J]. 沈阳航空航天大学学报, 2023, 40(4): 88-96 DOI:10.3969/j.issn.2095-1248.2023.04.012

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