  We introduce a domain decomposition-based nonlinear preconditioned iteration
for solving nonlinear, nonsmooth elliptic optimal control problems, with a
nonlinear reaction term, $L^1$ regularization and box constraints on the
control function. The method is obtained by applying semismooth Newton to the
fixed-point equation of the parallel optimized Schwarz iteration. As a proof of
concept, numerical experiments are performed on two subdomains, as well as on a
multi-subdomain test case. The results show that it is possible to obtain
substantial improvements in robustness and efficiency with the new method,
relative to semismooth Newton applied directly to the full optimization
problem, provided appropriate Robin parameters and a good continuation strategy
are chosen.
