求解垂直互补问题的一阶和二阶微分方程法

Methods for Solving First- and Second-Order Differential Equations for Vertical Complementarity Problems

摘要: 垂直互补问题是非线性互补问题的重要部分，本文采用微分方程法求解垂直互补问题。首先将垂直互补问题转化为变分不等式，然后直接利用投影算子建立微分方程系统，得到方程系统解轨迹的聚点，从而解决原问题及其解的存在性以及稳定性。本文主要内容有：第一部分引言介绍垂直互补问题、变分不等式问题的背景、研究现状；第二部分介绍相关基础知识；第三部分通过垂直互补问题转换为循环单调映射变分不等式，再利用投影算子方程分别建立一阶和二阶微分方程系统，证明原问题的解轨迹收敛，并在三元极小化优化问题中验证解的存在性和稳定性；第四部分通过具体实验验证本文方法，并不断调整参数直观比较在不同实验条件下对实验最终结果的影响。

Abstract: The vertical complementarity problem is an important part of the nonlinear complementarity problem, and the differential equation method is used to solve the problem of vertical complementarity. Firstly, the vertical complementarity problem is transformed into variational inequality, and then the differential equation system is directly established by using the projection operator, and the convergence point of the solution trajectory of the square system is obtained, so as to solve the original problem and the existence and stability of its solution. The main contents of this paper are as follows: the introduction of the first part introduces the background and research status of the vertical complementarity problem and the variational inequality problem; the second part introduces the basics of the vertical complementarity problem and the variational inequality problem; in the third part, the vertical complementarity problem is transformed into a cyclic monotonic mapping variational inequality, and then the first-order and second-order differential equation systems are established by using the projection operator equation to prove the convergence of the solution trajectory of the original problem, and the existence and stability of the solution are verified in the ternary minimization optimization problem. In the fourth part, the method is verified through specific experiments, and the parameters are continuously adjusted to visually compare the influence of different experimental conditions on the final results of the experiment.