A Parallel-in-Time Gradient-Type Method for Discrete Time Optimal Control Problems

This paper introduces and analyzes a new parallel-in-time gradient type method for the solution of convex linear-quadratic discrete-time optimal control (DTOC) problems. Each iteration of the classical gradient method requires the solution of the forward-in-time state equation followed by the solution of the backward-in-time adjoint equation to compute the gradient. To introduce parallelism, the time steps are split into N groups corresponding to time subintervals. At the time subinterval boundaries state and adjoint information from the previous iteration is used. On each time subinterval the forward-in-time state equation is solved, the backwardin-time adjoint equation is solved, gradient-type information is generated, and the control are updated. These computations can be performed in parallel across time subintervals. State and adjoint information at time subinterval boundaries is then exchanged with neighboring subintervals and the process is repeated. The resulting iteration can be interpreted as a so-called (2N−1)-part iteration scheme. Convergence of the new parallel-in-time gradient type method is proven for suitable step-sizes by showing that an associated block companion matrix has spectral radius less than one. The performance of the new method is demonstrated on a DTOC problem obtained from a discretization of a 3D parabolic optimal control problem. In this example nearly perfect speed-up is observed for moderate number of time subdomains. This speed-up due to time decomposition multiplies existing speed-up due to parallelization in the solution of state and adjoint equations.