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 00:00:00 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:: National Science Foundation (NSF); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

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

References (45)

Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: Stability and moving-horizon approximations Keerthi, S. S.; Gilbert, E. G. Journal of Optimization Theory and Applications, Vol. 57, Issue 2 https://doi.org/10.1007/bf00938540	journal	May 1988
A parallel Newton-type method for nonlinear model predictive control Deng, Haoyang; Ohtsuka, Toshiyuki Automatica, Vol. 109 https://doi.org/10.1016/j.automatica.2019.108560	journal	November 2019
A Parallel Decomposition Scheme for Solving Long-Horizon Optimal Control Problems Shin, Sungho; Faulwasser, Timm; Zanon, Mario 2019 IEEE 58th Conference on Decision and Control (CDC) https://doi.org/10.1109/cdc40024.2019.9030139	conference	December 2019
A flying inverted pendulum Hehn, Markus; D'Andrea, Raffaello 2011 IEEE International Conference on Robotics and Automation https://doi.org/10.1109/icra.2011.5980244	conference	May 2011
Exponentially Accurate Temporal Decomposition for Long-Horizon Linear-Quadratic Dynamic Optimization Xu, Wanting; Anitescu, Mihai SIAM Journal on Optimization, Vol. 28, Issue 3 https://doi.org/10.1137/16m1081993	journal	January 2018
Exponential Decay in the Sensitivity Analysis of Nonlinear Dynamic Programming Na, Sen; Anitescu, Mihai SIAM Journal on Optimization, Vol. 30, Issue 2 https://doi.org/10.1137/19m1265065	journal	January 2020
Perturbed Kuhn-Tucker points and rates of convergence for a class of nonlinear-programming algorithms Robinson, Stephen M. Mathematical Programming, Vol. 7, Issue 1 https://doi.org/10.1007/bf01585500	journal	December 1974
On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming Wächter, Andreas; Biegler, Lorenz T. Mathematical Programming, Vol. 106, Issue 1 https://doi.org/10.1007/s10107-004-0559-y	journal	April 2005
Global Convergence of ADMM in Nonconvex Nonsmooth Optimization Wang, Yu; Yin, Wotao; Zeng, Jinshan Journal of Scientific Computing, Vol. 78, Issue 1 https://doi.org/10.1007/s10915-018-0757-z	journal	June 2018
Solution of discrete-time optimal control problems on parallel computers Wright, Stephen J. Parallel Computing, Vol. 16, Issue 2-3 https://doi.org/10.1016/0167-8191(90)90060-m	journal	December 1990
A survey of industrial model predictive control technology Qin, S. Joe; Badgwell, Thomas A. Control Engineering Practice, Vol. 11, Issue 7, p. 733-764 https://doi.org/10.1016/S0967-0661(02)00186-7	journal	July 2003
A continuation/GMRES method for fast computation of nonlinear receding horizon control Ohtsuka, Toshiyuki Automatica, Vol. 40, Issue 4 https://doi.org/10.1016/j.automatica.2003.11.005	journal	April 2004
The advanced-step NMPC controller: Optimality, stability and robustness Zavala, Victor M.; Biegler, Lorenz T. Automatica, Vol. 45, Issue 1 https://doi.org/10.1016/j.automatica.2008.06.011	journal	January 2009
Parallel cyclic reduction decomposition for dynamic optimization problems Wan, Wei; Eason, John P.; Nicholson, Bethany Computers & Chemical Engineering, Vol. 120 https://doi.org/10.1016/j.compchemeng.2017.09.023	journal	January 2019
Benchmarking ADMM in nonconvex NLPs Rodriguez, Jose S.; Nicholson, Bethany; Laird, Carl Computers & Chemical Engineering, Vol. 119 https://doi.org/10.1016/j.compchemeng.2018.08.036	journal	November 2018
New Architectures for Hierarchical Predictive Control Zavala, Victor M. IFAC-PapersOnLine, Vol. 49, Issue 7 https://doi.org/10.1016/j.ifacol.2016.07.214	journal	January 2016
A Stochastic Dual Dynamic Programming Framework for Multiscale MPC Kumar, Ranjeet; Wenzel, Michael J.; Ellis, Matthew J. IFAC-PapersOnLine, Vol. 51, Issue 20 https://doi.org/10.1016/j.ifacol.2018.11.041	journal	January 2018
Graph-Based Modeling and Decomposition of Energy Infrastructures Shin, Sungho; Coffrin, Carleton; Sundar, Kaarthik IFAC-PapersOnLine, Vol. 54, Issue 3 https://doi.org/10.1016/j.ifacol.2021.08.322	journal	January 2021
Stochastic model predictive control for central HVAC plants Kumar, Ranjeet; Wenzel, Michael J.; ElBsat, Mohammad N. Journal of Process Control, Vol. 90 https://doi.org/10.1016/j.jprocont.2020.03.015	journal	June 2020
Temporal Decomposition Scheme for Nonlinear Multisite Production Planning and Distribution Models Jackson, Jennifer R.; Grossmann, Ignacio E. Industrial & Engineering Chemistry Research, Vol. 42, Issue 13 https://doi.org/10.1021/ie030070p	journal	May 2003
Application of Interior-Point Methods to Model Predictive Control Rao, C. V.; Wright, S. J.; Rawlings, J. B. Journal of Optimization Theory and Applications, Vol. 99, Issue 3 https://doi.org/10.1023/a:1021711402723	journal	December 1998
Nominal stability of real-time iteration scheme for nonlinear model predictive control Diehl, M.; Findeisen, R.; Bock, H. G. IEE Proceedings - Control Theory and Applications, Vol. 152, Issue 3 https://doi.org/10.1049/ip-cta:20040008	journal	May 2005
Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs Grüne, Lars; Schaller, Manuel; Schiela, Anton ESAIM: Control, Optimisation and Calculus of Variations, Vol. 27 https://doi.org/10.1051/cocv/2021030	journal	January 2021
Benchmarking large-scale distributed convex quadratic programming algorithms Kozma, Attila; Conte, Christian; Diehl, Moritz Optimization Methods and Software, Vol. 30, Issue 1 https://doi.org/10.1080/10556788.2014.911298	journal	May 2014
Non quadratic smooth model of fatigue for optimal fatigue-oriented individual pitch control Collet, D.; Alamir, M.; Domenico, D. Di Journal of Physics: Conference Series, Vol. 1618, Issue 2 https://doi.org/10.1088/1742-6596/1618/2/022004	journal	September 2020
Diffusing-Horizon Model Predictive Control Shin, Sungho; Zavala, Victor M. IEEE Transactions on Automatic Control, Vol. 68, Issue 1 https://doi.org/10.1109/TAC.2021.3137100	journal	January 2023
Efficient implementation of the Riccati recursion for solving linear-quadratic control problems Frison, Gianluca; Jorgensen, John Bagterp 2013 IEEE International Conference on Control Applications (CCA) https://doi.org/10.1109/cca.2013.6662901	conference	August 2013
Temporal Lagrangian decomposition of model predictive control for hybrid systems Beccuti, A. G.; Geyer, T.; Morari, M. 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601) https://doi.org/10.1109/cdc.2004.1428793	conference	January 2004
A parallel structure exploiting factorization algorithm with applications to Model Predictive Control Nielsen, Isak; Axehill, Daniel 2015 54th IEEE Conference on Decision and Control (CDC) https://doi.org/10.1109/cdc.2015.7402830	conference	December 2015
Nonlinear programming strategies on high-performance computers Kang, Jia; Chiang, Naiyuan; Laird, Carl D. 2015 54th IEEE Conference on Decision and Control (CDC) https://doi.org/10.1109/cdc.2015.7402938	conference	December 2015
Parallelizing LQR Computation Through Endpoint-Explicit Riccati Recursion Laine, Forrest; Tomlin, Claire 2019 IEEE 58th Conference on Decision and Control (CDC) https://doi.org/10.1109/cdc40024.2019.9029974	conference	December 2019
Distributed Receding Horizon Control of Dynamically Coupled Nonlinear Systems Dunbar, William B. IEEE Transactions on Automatic Control, Vol. 52, Issue 7 https://doi.org/10.1109/tac.2007.900828	journal	July 2007
An Augmented Lagrangian Filter Method for Real-Time Embedded Optimization Chiang, Nai-Yuan; Huang, Rui; Zavala, Victor M. IEEE Transactions on Automatic Control, Vol. 62, Issue 12 https://doi.org/10.1109/tac.2017.2694806	journal	December 2017
Superconvergence of Online Optimization for Model Predictive Control Na, Sen; Anitescu, Mihai IEEE Transactions on Automatic Control, Vol. 68, Issue 3 https://doi.org/10.1109/tac.2022.3223323	journal	March 2023
Predictive Active Steering Control for Autonomous Vehicle Systems Falcone, Paolo; Borrelli, Francesco; Asgari, Jahan IEEE Transactions on Control Systems Technology, Vol. 15, Issue 3 https://doi.org/10.1109/tcst.2007.894653	journal	May 2007
Neural-network predictive control for nonlinear dynamic systems with time-delay No authors listed IEEE Transactions on Neural Networks, Vol. 14, Issue 2 https://doi.org/10.1109/tnn.2003.809424	journal	March 2003
General nonlinear modal representation of large scale power systems Shanechi, H. M.; Pariz, N.; Vaahedi, E. IEEE Transactions on Power Systems, Vol. 18, Issue 3 https://doi.org/10.1109/tpwrs.2003.814883	journal	August 2003
Real-Time Nonlinear Optimization as a Generalized Equation Zavala, Victor M.; Anitescu, Mihai SIAM Journal on Control and Optimization, Vol. 48, Issue 8 https://doi.org/10.1137/090762634	journal	January 2010
Convergence Analysis of Alternating Direction Method of Multipliers for a Family of Nonconvex Problems Hong, Mingyi; Luo, Zhi-Quan; Razaviyayn, Meisam SIAM Journal on Optimization, Vol. 26, Issue 1 https://doi.org/10.1137/140990309	journal	January 2016
A Sparsity Preserving Convexification Procedure for Indefinite Quadratic Programs Arising in Direct Optimal Control Verschueren, Robin; Zanon, Mario; Quirynen, Rien SIAM Journal on Optimization, Vol. 27, Issue 3 https://doi.org/10.1137/16m1081543	journal	January 2017
Efficient Model Predictive Control for Parabolic PDEs with Goal Oriented Error Estimation Grüne, Lars; Schaller, Manuel; Schiela, Anton SIAM Journal on Scientific Computing, Vol. 44, Issue 1 https://doi.org/10.1137/20M1356324	journal	February 2022
A Real-Time Iteration Scheme for Nonlinear Optimization in Optimal Feedback Control Diehl, Moritz; Bock, Hans Georg; Schlöder, Johannes P. SIAM Journal on Control and Optimization, Vol. 43, Issue 5 https://doi.org/10.1137/s0363012902400713	journal	January 2005
Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers Boyd, Stephen; Parikh, Neal; Chu, Eric https://doi.org/10.1561/2200000016	book	January 2011
Handling Long Horizons in MPC: A Stochastic Programming Approach Kumar, Ranjeet; Wenzel, Michael J.; Ellis, Matthew J. 2018 Annual American Control Conference (ACC) https://doi.org/10.23919/acc.2018.8430780	conference	June 2018
An Ô (log N ) Parallel Algorithm for Newton Step Computation in Model Predictive Control Nielsen, Isak; Axehill, Daniel IFAC Proceedings Volumes, Vol. 47, Issue 3 https://doi.org/10.3182/20140824-6-za-1003.01577	journal	January 2014

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Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs Grüne, Lars; Schaller, Manuel; Schiela, Anton ESAIM: Control, Optimisation and Calculus of Variations, Vol. 27 https://doi.org/10.1051/cocv/2021030	journal	January 2021
Exponential Decay of Sensitivity in Graph-Structured Nonlinear Programs Shin, Sungho; Anitescu, Mihai; Zavala, Victor M. SIAM Journal on Optimization, Vol. 32, Issue 2 https://doi.org/10.1137/21m1391079	journal	May 2022