On the Convergence of Overlapping Schwarz Decomposition for Nonlinear Optimal Control

Na, Sen; Shin, Sungho; Anitescu, Mihai; Zavala, Victor M.

doi:10.1109/tac.2022.3194087

On the Convergence of Overlapping Schwarz Decomposition for Nonlinear Optimal Control

Journal Article · Tue Jul 26 04:00:00 EDT 2022 · IEEE Transactions on Automatic Control

DOI:https://doi.org/10.1109/tac.2022.3194087· OSTI ID:2333647

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Univ. of Chicago, IL (United States)
Univ. of Wisconsin, Madison, WI (United States)
Argonne National Laboratory (ANL), Argonne, IL (United States); Univ. of Chicago, IL (United States)
Univ. of Wisconsin, Madison, WI (United States); Argonne National Laboratory (ANL), Argonne, IL (United States)

Here, we study the convergence properties of an overlapping Schwarz decomposition algorithm for solving nonlinear optimal control problems (OCPs). The algorithm decomposes the time domain into a set of overlapping subdomains, and solves all subproblems defined over subdomains in parallel. The convergence is attained by updating primal-dual information at the boundaries of overlapping subdomains. We show that the algorithm exhibits local linear convergence, and that the convergence rate improves exponentially with the overlap size. We also establish global convergence results for a general quadratic programming, which enables the application of the Schwarz scheme inside second-order optimization algorithms (e.g., sequential quadratic programming). The theoretical foundation of our convergence analysis is a sensitivity result of nonlinear OCPs, which we call "exponential decay of sensitivity" (EDS). Intuitively, EDS states that the impact of perturbations at domain boundaries (i.e., initial and terminal time) on the solution decays exponentially as one moves into the domain. Here, we expand a previous analysis available in the literature by showing that EDS holds for both primal and dual solutions of nonlinear OCPs, under uniform second-order sufficient condition, controllability condition, and boundedness condition. We conduct experiments with a quadrotor motion planning problem and a partial differential equations (PDE) control problem to validate our theory, and show that the approach is significantly more efficient than alternating direction method of multipliers and as efficient as the centralized interior-point solver.

View Accepted Manuscript (DOE)

Research Organization:: Argonne National Laboratory (ANL), Argonne, IL (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF)

Grant/Contract Number:: AC02-06CH11357

OSTI ID:: 2333647

Journal Information:: IEEE Transactions on Automatic Control, Journal Name: IEEE Transactions on Automatic Control Journal Issue: 11 Vol. 67; ISSN 0018-9286

Publisher:: IEEECopyright Statement

Country of Publication:: United States

Language:: English

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