Nonlinear preconditioned primal-dual method for a class of structured minimax problems
Lu Zhang, Hongxia Wang, Hui Zhang
TL;DR
This work addresses nonconvex-nonconcave, non-smooth saddle-point problems by introducing a nonlinear, potentially asymmetric preconditioned primal-dual method with projection. The core idea combines a warped resolvent-based preconditioner with a separating-hyperplane projection to guarantee convergence despite nonlinearities, and it yields weak convergence, sublinear convergence under convexity, and linear convergence under metric subregularity. The framework unifies many existing primal-dual algorithms as special cases and provides explicit conditions for well-defined resolvents and parameter choices. The approach offers a flexible toolkit for structured saddle-point problems across applications, with clear pathways for future enhancements of the preconditioner and correction mechanisms.
Abstract
We propose and analyze a general framework called nonlinear preconditioned primal-dual with projection for solving nonconvex-nonconcave and non-smooth saddle-point problems. The framework consists of two steps. The first is a nonlinear preconditioned map followed by a relaxed projection onto the separating hyperspace we construct. One key to the method is the selection of preconditioned operators, which tailors to the structure of the saddle-point problem and is allowed to be nonlinear and asymmetric. The other is the construction of separating hyperspace, which guarantees fast convergence. This framework paves the way for constructing nonlinear preconditioned primal-dual algorithms. We show that weak convergence, and so is sublinear convergence under the assumption of the convexity of saddle-point problems and linear convergence under a metric subregularity. We also show that many existing primal-daul methods, such as the generalized primal-dual algorithm method, are special cases of relaxed preconditioned primal-dual with projection.