带扰动项的阻尼惯性梯度系统求解二阶锥约束变分不等式
Damped Inertial Gradient System with a Perturbation Term for Solving Second-Order Cone Constrained Variational Inequalities
摘要: 本文运用光滑化的自然残差函数建立了二阶锥约束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带扰动项的阻尼惯性梯度系统求解该无约束优化问题，并证明了该系统的稳定性，从而得到了二阶锥约束变分不等式问题的KKT点的收敛性。并将带扰动项的阻尼惯性梯度系统与已有的一阶微分方程系统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，带扰动项的阻尼惯性梯度系统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一些，但是差距不大。
Abstract: In this paper, a smoothed KKT system of equations for the second-order cone constrained variational inequality problem is established by using the smoothed natural residual function, and an equivalent unconstrained optimization problem is constructed. A damped inertial gradient system with a perturbation term, which includes damped inertial parameters and time scale parameters, is developed to solve this unconstrained optimization problem. The stability of this system is proved, thus obtaining the convergence of KKT points for the second-order cone constrained variational inequality problem. Furthermore, the damped inertial gradient system with a perturbation term is compared with the existing first-order differential equation system methods in terms of theoretical conditions and numerical results. In terms of theoretical condition requirements, the conditions of the damped inertial gradient system with a perturbation term are easier to implement; in terms of numerical results, the first-order differential equation method has a slightly faster convergence speed, but the gap is not significant.
文章引用：杨帆, 郑慧心, 胡钰婷, 王莉. 带扰动项的阻尼惯性梯度系统求解二阶锥约束变分不等式[J]. 理论数学, 2025, 15(9): 136-145. https://doi.org/10.12677/pm.2025.159241

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