  This paper presents a novel backtracking strategy for additive Schwarz
methods for general convex optimization problems as an acceleration scheme. The
proposed backtracking strategy is independent of local solvers, so that it can
be applied to any algorithms that can be represented in an abstract framework
of additive Schwarz methods. Allowing for adaptive increasing and decreasing of
the step size along the iterations, the convergence rate of an algorithm is
greatly improved. Improved convergence rate of the algorithm is proven
rigorously. In addition, combining the proposed backtracking strategy with a
momentum acceleration technique, we propose a further accelerated additive
Schwarz method. Numerical results for various convex optimization problems that
support our theory are presented.
