二阶锥约束变分不等式的阻尼惯性梯度系统的方法研究
Research on the Method of Damped Inertial Gradient System for Second-Order Cone Constrained Variational Inequalities
摘要: 本文聚焦二阶锥约束变分不等式问题，旨在探寻有效的求解方法与分析其相关性质。首先，通过分析KKT条件并借助光滑化的二阶锥互补函数，将二阶锥约束变分不等式问题转化为无约束优化问题。随后，受Attouch等工作启发，构造阻尼惯性梯度系统来求解该无约束优化问题。在此基础上，对阻尼惯性梯度系统进行稳定性分析，在给定的阻尼系数、时间尺度系数条件及特定增长条件下，证明了系统解的轨迹收敛速度和有界性。研究成果为二阶锥约束变分不等式问题的求解提供了新的思路和理论依据。
Abstract: This paper focuses on the second-order cone constrained variational inequalities problem, aiming to explore effective solution methods and analyze its relevant properties. Firstly, by analyzing the KKT conditions and using the smoothed second-order cone complementarity function, the second-order cone constrained variational inequality problem is transformed into an unconstrained optimization problem. Subsequently, inspired by the work of Attouch et al., a damped inertial gradient system is constructed to solve this unconstrained optimization problem. On this basis, the stability analysis of the damped inertial gradient system is conducted. Under the given conditions of damping coefficients, time scale coefficients, and specific growth conditions, the convergence rate and boundedness of the trajectories of the system’s solutions are proved. The research results provide new ideas and theoretical basis for solving the second-order cone constrained variational inequality problem.
文章引用：宋瑷如, 于晓雪, 施玉缘, 王莉. 二阶锥约束变分不等式的阻尼惯性梯度系统的方法研究[J]. 理论数学, 2025, 15(9): 165-174. https://doi.org/10.12677/pm.2025.159243

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