  We demonstrate that difficult non-convex non-smooth optimization problems,
such as Nash equilibrium problems and anisotropic as well as isotropic Potts
segmentation model, can be written in terms of generalized conjugates of convex
functionals. These, in turn, can be formulated as saddle-point problems
involving convex non-smooth functionals and a general smooth but non-bilinear
coupling term. We then show through detailed convergence analysis that a
conceptually straightforward extension of the primal--dual proximal splitting
method of Chambolle and Pock is applicable to the solution of such problems.
Under sufficient local strong convexity assumptions of the functionals -- but
still with a non-bilinear coupling term -- we even demonstrate local linear
convergence of the method. We illustrate these theoretical results numerically
on the aforementioned example problems.
