  We develop multigrid-in-time preconditioners for Karush-Kuhn-Tucker (KKT)
systems that arise in the solution of time-dependent optimization problems. We
focus on a specific instance of KKT systems, known as augmented systems, which
underpin the composite-step sequential quadratic programming framework [1]. To
enable time-domain decomposition, our approach introduces virtual state
variables and continuity constraints at each discrete time interval. The
virtual state variables not only facilitate a decoupling in time but also give
rise to fixed-point iterations that aid the solution of KKT systems. These
fixed-point schemes can be used either as preconditioners for Krylov subspace
methods or as smoothers for multigrid-in-time schemes. For the latter, we
develop a block-Jacobi scheme that parallelizes trivially in the time domain.
To complete the multigrid construction, we use simple prolongation and
restriction operators based on geometric multigrid ideas, and a coarse-grid
solver based on a GMRES iteration preconditioned with the symmetric block
Gauss-Seidel scheme. We present two optimal control examples, involving the
viscous Burgers' and van der Pol oscillator equations, respectively, and
demonstrate algorithmic scalability.
