1) I assume that you are talking about TGVDenoise. It uses the Chambolle algorithm, where the minimization problem is expanded with the dual to a concave-convex problem, and a saddle point is searched.
Chambolle, A., Pock, T. A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120–145, 2011.