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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Bibliographic Details
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
Online Access:Get full text
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Summary: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.
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AC02-06CH11357
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
National Science Foundation (NSF)
ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2022.3194087