  This paper gives a unified convergence analysis of additive Schwarz methods
for general convex optimization problems. Resembling to the fact that additive
Schwarz methods for linear problems are preconditioned Richardson methods, we
prove that additive Schwarz methods for general convex optimization are in fact
gradient methods. Then an abstract framework for convergence analysis of
additive Schwarz methods is proposed. The proposed framework applied to linear
elliptic problems agrees with the classical theory. We present applications of
the proposed framework to various interesting convex optimization problems such
as nonlinear elliptic problems, nonsmooth problems, and nonsharp problems.
