  The alternating direction method of multipliers (ADMM) algorithm is a
powerful and flexible tool for complex optimization problems of the form
$\min\{f(x)+g(y) : Ax+By=c\}$. ADMM exhibits robust empirical performance
across a range of challenging settings including nonsmoothness and nonconvexity
of the objective functions $f$ and $g$, and provides a simple and natural
approach to the inverse problem of image reconstruction for computed tomography
(CT) imaging. From the theoretical point of view, existing results for
convergence in the nonconvex setting generally assume smoothness in at least
one of the component functions in the objective. In this work, our new
theoretical results provide convergence guarantees under a restricted strong
convexity assumption without requiring smoothness or differentiability, while
still allowing differentiable terms to be treated approximately if needed. We
validate these theoretical results empirically, with a simulated example where
both $f$ and $g$ are nondifferentiable -- and thus outside the scope of
existing theory -- as well as a simulated CT image reconstruction problem.
