Published

Transformed Primal-Dual Methods for Nonlinear Saddle Point Systems

Long Chen and Jingrong Wei

Journal of Numerical Mathematics. January 15, 2023.

arXiv   Bibtex coming soon

ABSTRACT:

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.