带扰动项的二阶微分方程方法求解变分不等式问题

贾丹娜; 王莉; 孙菊贺; 庄慧婷; 袁艳红

doi:10.3969/j.issn.2095-1248.2023.05.012

沈阳航空航天大学学报 ›› 2023, Vol. 40 ›› Issue (5) : 90 -96. DOI: 10.3969/j.issn.2095-1248.2023.05.012

基础科学和工程

带扰动项的二阶微分方程方法求解变分不等式问题

贾丹娜 ¹ ,
王莉 ¹ ,
孙菊贺 ¹ ,
庄慧婷 ¹ ,
袁艳红 ²

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

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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。

Abstract

The second-order differential equations with perturbation terms to solve variational inequality problem were focused on and discussed the convergence of its solution and the speed of the convergence. Firstly，the Karush-Kuhn-Tucker （KKT） conditions of the original variational inequality problem were equivalently transformed into a system of smoothing equations by using a smoothing complementary 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 final 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 convergence 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.