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, and implicit-explicit methods 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:用于非线性平滑马鞍系统的原始双向( TPD) 流为非线性平滑马鞍系统类别。 双变量的流中包含一个非常精细的Schur 补充。 通过显示强大的 Lyapunov 属性, 能够取得马鞍点的指数稳定性。 几个TPD 迭代来自 TPD 流中隐含的 Euler 、 直线 Euler 和隐含的表达方法 。 概括为对称 TPD 迭代, 线性趋同率为对等式 TPD 支架点系统保留了线性趋同率, 假设正统性功能是很强的正弦。 增强的拉格朗格方法的有效性可以解释为非强性凝结的正规化和Schur 补充的前提条件。 算法和趋同性分析关键地取决于原始变量和双重变量空间的适当内部产品。 还开发了与非线性内溶剂的明确的趋同性分析。