一类解决可分离优化问题动力系统的渐进行为分析

A Gradual Asymptotic Behavior Analysis of Dynamical Systems for Solving Separable Optimization Problems

摘要: 本文提出了一类关于二阶原始变量和一阶对偶变量组合的惯性原对偶动力系统，利用该惯性原对 偶动力系统来解决用来解决一类可分离的优化问题。 我们的方法与传统的优化算法不同，该方法 利用连续性的方程来解决最优化问题，在具体的应用过程当中，能够使最目标函数快速且渐近的 收敛到最优解。 我们主要得到两个重要理论：一方面在解决凸问题时，得到了该系统解的存在唯 一性；另一方面，基于Lyapunov分析方法的应用，在得到目标函数收敛到最优解的同时，还得到 了其收敛的速率为O(1/t²), 这也是己知的关于动力系统最好的收敛性结果。

Abstract: This paper introduces a class of inertial primal-dual dynamical systems involving second-order primal variables and first-order dual variables, which are used to solve a class of separable optimization problems. Our approach differs from traditional op- timization algorithms in that it utilizes continuous equations to solve optimization problems, enabling the objective function to converge quickly and asymptotically to the optimal solution in practical applications. We have obtained two important the- oretical results: on the one hand, the system has a unique solution; on the other hand, based on the application of the Lyapunov analysis method, we have not only obtained the convergence of the objective function to the optimal solution but also achieved a convergence rate of O(1/t²), which is also the best known convergence result for dynamical systems.