惯性原对偶动力系统在优化中的应用  

Application of Inertial Primal-Dual Dynamical Systems in Optimization

作　　者：乔立凯 

机构地区：[1]河北工业大学理学院,天津

出　　处：《应用数学进展》2024年第12期5237-5251,共15页Advances in Applied Mathematics

摘　　要：本文提出了一类带有Tikhonov正则项的惯性原对偶动力系统,将该系统与最优化算法建立联 系,用来解决一类带有等式约束的优化问题。 我们的方法利用Tikhonov正则项以及惯性技术,将 己有的传统惯性动力系统进行改进,主要得到两个重要理论:一方面通过对相应的优化问题求 解,在求解的具体过程当中,得到了最优化理论中目标函数是收敛到最优值的;另一方面,基 于Tikhonov正则项技术的应用,在得到目标函数收敛性的同时,还得到了轨道(即最优解)的强 收敛性,这也是到目前为止,关于动力系统最好的结果。 我们提出的动力系统能够解决带有等式 约束的最优化问题,并且轨道可以强收敛到全局解的最小范数元素。This paper proposes a class of inertial primal-dual dynamic systems with a Tikhonov regularization term, establishing a connection between this system and optimization algorithms to address a class of optimization problems with equality constraints. Our method utilizes the Tikhonov regularization term and inertial techniques to improve existing traditional inertial dynamic systems, leading to two important theoretical re- sults: on one hand, by solving the corresponding optimization problem, we achieve convergence of the objective function to its optimal value during the specific solution process;on the other hand, based on the application of the Tikhonov regularization technique, we obtain not only convergence of the objective function but also strong convergence of the trajectory (i.e., the optimal solution), which represents the best result regarding dynamic systems to date. Our proposed dynamic system can solve optimization problems with equality constraints, and the trajectory can strongly con- verge to the element of minimum norm in the global solution.

关 键 词：动力系统 最优化理论 Laypnuov分析 强收敛 

分 类 号：O17[理学—数学]