一类变分不等式问题的二阶微分方程方法  

The Second Order Differential Equation Method for a Class of Variational Inequality Problem

作　　者：于晓雪 安思源 宋瑷如 施玉缘 唐博文 陈宇航 王莉 

机构地区：[1]沈阳航空航天大学理学院,辽宁 沈阳

出　　处：《应用数学进展》2024年第12期5439-5449,共11页Advances in Applied Mathematics

摘　　要：本文运用光滑化的自然残差函数建立了具有不等式约束条件的变分不等式问题的光滑化KKT方程组,并建立了与其等价的无约束优化问题。建立了具有阻尼惯性参数和时间尺度参数的二阶微分方程系统来求解该无约束优化问题,并证明了该二阶微分方程系统的稳定性,从而得到了具有不等式约束的变分不等式问题的KKT点的收敛性。并将二阶微分方程方法与已有的一阶微分方程方法进行了理论条件和数值结果的对比。在理论条件的要求上,二阶微分方程方法的条件要更容易实现,而在数值结果上,一阶微分方程方法的收敛速度要快,但是两种方法的差距可以忽略不计。In this paper, a smoothing KKT equation system with inequality constraints is established by using the smoothing natural residual function, and the unconstrained optimization problem is established. A second order differential equation system with damping coefficient and time scale coefficient is established to solve the unconstrained optimization problem, and the stability of the second-order differential equation system is proved, then the convergence of the KKT points of the variational inequality problem with inequality constraints is obtained. The theoretical conditions and numerical results of the second-order differential equation method are compared with the existing first order differential equation method. In terms of theoretical conditions, the conditions of the second-order differential equation method are easier to implement, while in the numerical results, the convergence speed of the first order differential equation method is faster, but the difference between the two methods is negligible.

关 键 词：变分不等式 二阶微分方程方法 无约束优化 阻尼惯性参数 时间尺度参数 

分 类 号：O17[理学—数学]