Nonlinear preconditioned primal-dual method for a class of structured minimax problems

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.