A transformed primal-dual (TPD) flow is developed for a class of nonlinear smooth saddle point system. The flow for the dual variable contains a Schur complement which is strongly convex. Exponential stability of the saddle point is obtained by showing the strong Lyapunov property. Several TPD iterations are derived by implicit Euler, explicit Euler, implicit-explicit and Gauss-Seidel methods with accelerated overrelaxation of the TPD flow. Generalized to the symmetric TPD iterations, linear convergence rate is preserved for convex-concave saddle point systems under assumptions that the regularized functions are strongly convex. The effectiveness of augmented Lagrangian methods can be explained as a regularization of the non-strongly convexity and a preconditioning for the Schur complement. The algorithm and convergence analysis depends crucially on appropriate inner products of the spaces for the primal variable and dual variable. A clear convergence analysis with nonlinear inexact inner solvers is also developed.

翻译：针对一类非线性光滑鞍点系统，提出了一种转换原始-对偶(TPD)流。该对偶变量流包含一个强凸的舒尔补。通过证明强李雅普诺夫性质，实现了鞍点的指数稳定性。基于隐式欧拉、显式欧拉、隐式-显式及高斯-赛德尔方法，结合TPD流的加速超松弛技术，推导出若干TPD迭代格式。推广至对称TPD迭代后，在正则化函数强凸的假设条件下，凸-凹鞍点系统仍保持线性收敛速率。增广拉格朗日方法的有效性可解释为对非强凸性的正则化及对舒尔补的预处理。算法设计与收敛性分析关键依赖于原始变量与对偶变量空间的适当内积。本文还建立了含非线性非精确内迭代求解器的清晰收敛性分析框架。