A forward-backward splitting algorithm for the minimization of non-smooth convex functionals in Banach space
This work provides a theoretical foundation for optimization in Banach spaces, which is important for problems like total-variation regularization, but the results are incremental as they generalize existing Hilbert space methods.
The authors propose a forward-backward splitting algorithm for minimizing sums of smooth and non-smooth convex functionals in Banach spaces, extending the Hilbert space method. They prove descent and convergence properties, obtain convergence rates via Bregman-Taylor-distance estimates, and demonstrate applications to inverse problems and image restoration.
We consider the task of computing an approximate minimizer of the sum of a smooth and non-smooth convex functional, respectively, in Banach space. Motivated by the classical forward-backward splitting method for the subgradients in Hilbert space, we propose a generalization which involves the iterative solution of simpler subproblems. Descent and convergence properties of this new algorithm are studied. Furthermore, the results are applied to the minimization of Tikhonov-functionals associated with linear inverse problems and semi-norm penalization in Banach spaces. With the help of Bregman-Taylor-distance estimates, rates of convergence for the forward-backward splitting procedure are obtained. Examples which demonstrate the applicability are given, in particular, a generalization of the iterative soft-thresholding method by Daubechies, Defrise and De Mol to Banach spaces as well as total-variation based image restoration in higher dimensions are presented.