The primal-dual hybrid gradient method for semiconvex splittings

This paper deals with the analysis of a recent reformulation of the primal-dual hybrid gradient method, which allows one to apply it to nonconvex regularizers. Particularly, it investigates variational problems for which the energy to be minimized can be written as G(u) + F(Ku), where G is convex, F is semiconvex, and K is a linear operator. We study the method and prove convergence in the case where the nonconvexity of F is compensated for by the strong convexity of G. The convergence proof yields an interesting requirement for the choice of algorithm parameters, which we show to be not only sufficient, but also necessary. Additionally, we show boundedness of the iterates under much weaker conditions. Finally, in several numerical experiments we demonstrate effectiveness and convergence of the algorithm beyond the theoretical guarantees.