  Fourth-order variational inequalities are encountered in various scientific
and engineering disciplines, including elliptic optimal control problems and
plate obstacle problems. In this paper, we consider additive Schwarz methods
for solving fourth-order variational inequalities. Based on a unified framework
of various finite element methods for fourth-order variational inequalities, we
develop one- and two-level additive Schwarz methods. We prove that the
two-level method is scalable in the sense that the convergence rate of the
method depends on $H/h$ and $H/\delta$ only, where $h$ and $H$ are the typical
diameters of an element and a subdomain, respectively, and $\delta$ measures
the overlap among the subdomains. This proof relies on a new nonlinear
positivity-preserving coarse interpolation operator, the construction of which
was previously unknown. To the best of our knowledge, this analysis represents
the first investigation into the scalability of the two-level additive Schwarz
method for fourth-order variational inequalities. Our theoretical results are
verified by numerical experiments.
