On the Convergence of Overlapping Schwarz Decomposition for Nonlinear Optimal Control

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...

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Published in:	IEEE transactions on automatic control Vol. 67; no. 11; pp. 5996 - 6011
Main Authors:	Na, Sen, Shin, Sungho, Anitescu, Mihai, Zavala, Victor M.
Format:	Journal Article
Language:	English
Published:	New York IEEE 01.11.2022 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:	Algorithms Boundaries Convergence Decomposition Decomposition methods ENGINEERING Motion planning Nonlinear control nonlinear programming Optimal control Optimization overlapping parallel algorithms Partial differential equations Perturbation Perturbation methods Prediction algorithms Predictive control Quadratic programming Sensitivity Sensitivity analysis Trajectory
ISSN:	0018-9286, 1558-2523
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Abstract	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.
AbstractList	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. 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.
Author	Shin, Sungho Na, Sen Anitescu, Mihai Zavala, Victor M.
Author_xml	– sequence: 1 givenname: Sen orcidid: 0000-0002-7977-5276 surname: Na fullname: Na, Sen email: senna@uchicago.edu organization: Department of Statistics, University of Chicago, Chicago, IL, USA – sequence: 2 givenname: Sungho orcidid: 0000-0002-9889-3278 surname: Shin fullname: Shin, Sungho email: sungho.shin@wisc.edu organization: Department of Chemical and Biological Engineering, University of Wisconsin-Madison, Madison, WI, USA – sequence: 3 givenname: Mihai orcidid: 0000-0002-0787-5462 surname: Anitescu fullname: Anitescu, Mihai email: anitescu@mcs.anl.gov organization: Mathematics and Computer Science Division, Argonne National Laboratory, Lemont, IL, USA – sequence: 4 givenname: Victor M. orcidid: 0000-0002-5744-7378 surname: Zavala fullname: Zavala, Victor M. email: victor.zavala@wisc.edu organization: Department of Chemical and Biological Engineering, University of Wisconsin-Madison, Madison, WI, USA
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Snippet	We study the convergence properties of an overlapping Schwarz decomposition algorithm for solving nonlinear optimal control problems (OCPs). The algorithm... Here, we study the convergence properties of an overlapping Schwarz decomposition algorithm for solving nonlinear optimal control problems (OCPs). The...
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SubjectTerms	Algorithms Boundaries Convergence Decomposition Decomposition methods ENGINEERING Motion planning Nonlinear control nonlinear programming Optimal control Optimization overlapping parallel algorithms Partial differential equations Perturbation Perturbation methods Prediction algorithms Predictive control Quadratic programming Sensitivity Sensitivity analysis Trajectory
Title	On the Convergence of Overlapping Schwarz Decomposition for Nonlinear Optimal Control
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