Precompact convergence of the nonconvex Primal-Dual Hybrid Gradient algorithm

The Primal-Dual Hybrid Gradient (PDHG) algorithm is a powerful algorithm used quite frequently in recent years for solving saddle-point optimization problems. The classical application considers convex functions, and it is well studied in literature. In this paper, we consider the convergence of an alternative formulation of the PDHG algorithm in the nonconvex case under the precompact assumption. The proofs are based on the Kurdyka-Ł ojasiewic functions, that cover a wide range of problems. A simple numerical experiment illustrates the convergence properties.