A Parallel Decomposition Scheme for Solving Long-Horizon Optimal Control Problems

Abstract

We present a temporal decomposition scheme for solving long-horizon optimal control problems. In the proposed scheme, the time domain is decomposed into a set of subdomains with partially overlapping regions. Subproblems associated with the subdomains are solved in parallel to obtain local primal-dual trajectories that are assembled to obtain the global trajectories. We provide a sufficient condition that guarantees convergence of the proposed scheme. This condition states that the effect of perturbations on the boundary conditions (i.e., the initial state and terminal dual/adjoint variable) should decay asymptotically as one moves away from the boundaries. This condition also reveals that the scheme converges if the size of the overlap is sufficiently large and that the convergence rate improves with the size of the overlap. We prove that linear quadratic problems satisfy the asymptotic decay condition, and we discuss numerical strategies to determine if the condition holds in more general cases. We draw upon a non-convex optimal control problem to illustrate the performance of the proposed scheme.

Temporal Parallelization of Dynamic Programming and Linear Quadratic Control

“…Definition 3: Given elements a and b of the form (19), the binary associative operator for dynamic programming is…”

“…An iterated method for linear quadratic control problems, with constraints, in which each step can be parallelised is proposed in [18], though it requires positive definite matrices in the cost function and may require regularisation. An algorithm to approximately solve an optimal control problem by solving different subproblems with partially overlapping time windows is provided in [19], and an approximate parallel algorithm for linear MPC is given in [20]. Reference [21] provides combination rules to separate the dynamic programming algorithm into different subproblems across the temporal domain.…”

A Novel Macro and Micro Scheduling Model for Regulated Safety Inspections Based on Probability Integral Transform and Modified Bin Packing Algorithm

“…Then, the small problems can be solved by relatively simple techniques since the search spaces of the sub-problems are significantly smaller than that of the original one [56]. This approach was applied to solve timetabling problems such as examination timetabling [44], crew scheduling [50], long term operational scheduling of cogeneration energy systems [57], information macro and micro scheduling in wireless communication systems [58], water resources macro and micro scheduling [59] and other applications in many fields of logistics, chemical, process and energy industries [60].…”

A Fast Temporal Decomposition Procedure for Long-horizon Nonlinear Dynamic Programming

“…As these solvers neglect to take advantage of generic properties of dynamical systems, they are deficient when directly implemented on Problem (1.1), especially when we have limited computing resources. Generic properties of dynamical systems, such as sensitivity and controllability, play a key role in developing various efficient algorithms [48,25,38,39,40]. Furthermore, the centralized solvers are not flexible enough to be implemented on different types of computing hardware.…”