论文标题
转换的非线性鞍点系统的原始偶对偶方法
Transformed Primal-Dual Methods For Nonlinear Saddle Point Systems
论文作者
论文摘要
为一类非线性平滑鞍点系统开发了转换的原始偶（TPD）流。双重变量的流量包含一个强烈凸的Schur补体。鞍点的指数稳定性是通过显示强的Lyapunov特性获得的。有几种TPD迭代是由隐式Euler，显式Euler，隐式解释和高斯 - 西德尔方法得出的，并且加速了TPD流的过度递延。在对称的TPD迭代中概括，在正则化功能强烈凸的假设下，在凸 - 孔符号鞍点系统中保留了线性收敛速率。增强拉格朗日方法的有效性可以解释为非局部凸度的正规化和Schur补体的预处理。算法和收敛分析至关重要地取决于原始变量和偶变量的空间的适当内部产物。还开发了针对非线性不精性内求解器的明确收敛分析。
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.
