带扰动项的二阶微分方程方法求解变分不等式问题  

The second⁃order differential equation method with perturbation terms for solving variational inequality problem

作　　者：贾丹娜 王莉 孙菊贺 庄慧婷 袁艳红[2] JIA Danna;WANG Li;SUN Juhe;ZHUANG Huiting;YUAN Yanhong(College of Science,Shenyang Aerospace University,Shenyang 110136,China;College of Economic and Management,Taiyuan University of Technology,Taiyuan 030024,China)

机构地区：[1]沈阳航空航天大学理学院,沈阳110136 [2]太原理工大学经济管理学院,太原030024

出　　处：《沈阳航空航天大学学报》2023年第5期90-96,共7页Journal of Shenyang Aerospace University

基　　金：国家自然科学基金(项目编号:11901422)。

摘　　要：利用带扰动项的二阶微分方程方法求解变分不等式问题,并讨论其解的收敛性和收敛速度。首先,通过对原始变分不等式问题所对应的Karush-Kuhn-Tucker(KKT)条件进行等价转换后,借助光滑化的互补函数,等价转化成求解光滑方程组S(ε,x,μ,λ)=0,进一步等价于求解一个无约束优化问题;其次,建立带扰动项的二阶微分方程系统来求解最终的无约束优化问题,并在一定的约束条件下,得到了该二阶微分方程系统的解稳定性及收敛速度,即得到了所求的变分不等式问题的收敛性和解的收敛速度;最后,给出数值实验说明所提出的微分方程方法求解变分不等式的有效性。The second-order differential equations with perturbation terms to solve variational inequali-ty problem were focused on and discussed the convergence of its solution and the speed of the conver-gence.Firstly,the Karush-Kuhn-Tucker(KKT)conditions of the original variational inequality prob-lem were equivalently transformed into a system of smoothing equations by using a smoothing comple-mentary function,and it was furtherly equivalent to an unconstrained optimization problem.Secondly,a system of second-order differential equations with perturbation terms was established to solve the fi-nal unconstrained optimization problem and discuss the stability of the differential equation system and the speed of convergence under the certain conditions.The convergence and the speed of the conver-gence for the solution to the original variational inequality problem was discussed.Finally,numerical experiments were given to show the effectiveness of the differential equation method for solving the variational inequality problem.

关 键 词：变分不等式 二阶微分方程 扰动项 Karush-Kuhn-Tucker条件 稳定性 

分 类 号：O224[理学—运筹学与控制论]