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.

中文翻译：

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非线性鞍点系统的变换原始对偶方法

为一类非线性光滑鞍点系统开发了变换的原始对偶 (TPD) 流。对偶变量的流包含一个强凸的 Schur 补集。鞍点的指数稳定性是通过显示强李雅普诺夫性质获得的。几个 TPD 迭代是通过隐式 Euler、显式 Euler、隐式-显式和 Gauss-Seidel 方法导出的，并加速了 TPD 流的过度松弛。推广到对称 TPD 迭代，假设正则化函数是强凸的，凸凹鞍点系统保持线性收敛率。增广拉格朗日方法的有效性可以解释为非强凸性的正则化和 Schur 补集的预处理。算法和收敛性分析关键取决于原始变量和对偶变量空间的适当内积。还开发了具有非线性不精确内部求解器的清晰收敛分析。