  For a Hilbert space setting Chambolle and Pock introduced an attractive
first-order algorithm which solves a convex optimization problem and its
Fenchel dual simultaneously. We present a generalization of this algorithm to
Banach spaces. Moreover, under certain conditions we prove strong convergence
as well as convergence rates. Due to the generalization the method becomes
efficiently applicable for a wider class of problems. This fact makes it
particularly interesting for solving ill-posed inverse problems on Banach
spaces by Tikhonov regularization or the iteratively regularized Newton-type
method, respectively.
