Overlapping Schwarz Scheme for Linear-Quadratic Programs in Continuous Time

References

• [1] Anderson BD, Moore JB (2007) Optimal Control: Linear Quadratic Methods (Courier Corporation, Chelmsford, MA).Google Scholar
• [2] Antil H, Leykekhman D (2018) A brief introduction to PDE-constrained optimization. Antil H, Kouri DP, Lacasse MD, Ridzal D, eds. Frontiers in PDE-Constrained Optimization (Springer, New York), 3–40.Crossref, Google Scholar
• [3] Athans M, Falb PL (2013) Optimal Control: An Introduction to the Theory and Its Applications (Courier Corporation, Chelmsford, MA).Google Scholar
• [4] Barron EN, Jensen R (1986) The Pontryagin maximum principle from dynamic programming and viscosity solutions to first-order partial differential equations. Trans. Amer. Math. Soc. 298(2):635–641.Crossref, Google Scholar
• [5] Bertsekas DP (2005) Dynamic Programming and Optimal Control, vol. I, 3rd ed. (Athena Scientific, Belmont, MA).Google Scholar
• [6] Betts JT (2010) Practical Methods for Optimal Control and Estimation Using Nonlinear Programming (Society for Industrial and Applied Mathematics, Philadelphia).Crossref, Google Scholar
• [7] Bock H, Plitt K (1984) A multiple shooting algorithm for direct solution of optimal control problems. IFAC Proc. Vol. 17(2):1603–1608.Crossref, Google Scholar
• [8] Bokanowski O, Désilles A, Zidani H (2021) Relationship between maximum principle and dynamic programming in presence of intermediate and final state constraints. ESAIM COCV 27:91.Crossref, Google Scholar
• [9] Borkar VS (2024) Contributions to the theory of optimal control. Resonance 29(9):1309–1327.Crossref, Google Scholar
• [10] Brenner SC, Scott LR (2008) The Mathematical Theory of Finite Element Methods (Springer, New York).Crossref, Google Scholar
• [11] Chachuat B (2007) Nonlinear and dynamic optimization: From theory to practice. Teaching Material. https://infoscience.epfl.ch/handle/20.500.14299/12332.Google Scholar
• [12] Craig W, ed. (2008) Hamiltonian Dynamical Systems and Applications (Springer Netherlands, Dordrecht, Netherlands).Crossref, Google Scholar
• [13] Cuomo S, Di Cola VS, Giampaolo F, Rozza G, Raissi M, Piccialli F (2022) Scientific machine learning through physics-informed neural networks: Where we are and what’s next. J. Sci. Comput. 92(3):88. Crossref, Google Scholar
• [14] Damm T, Grüne L, Stieler M, Worthmann K (2014) An exponential turnpike theorem for dissipative discrete time optimal control problems. SIAM J. Control Optim. 52(3):1935–1957.Crossref, Google Scholar
• [15] De los Reyes JC (2015) Numerical PDE-Constrained Optimization (Springer International Publishing, Cham, Switzerland).Crossref, Google Scholar
• [16] Dinh S, Tong Y, Wei Z, Gerdes O, Biegler L (2025) Nonlinear model predictive control with an infinite horizon approximation. J. Process Control. 155:103565.Crossref, Google Scholar
• [17] Dolean V, Jolivet P, Nataf F (2015) An Introduction to Domain Decomposition Methods: Algorithms, Theory, and Parallel Implementation (Society for Industrial and Applied Mathematics, Philadelphia).Crossref, Google Scholar
• [18] Eichmeir P, Lauβ T, Oberpeilsteiner S, Nachbagauer K, Steiner W (2020) The adjoint method for time-optimal control problems. J. Comput. Nonlinear Dynam. 16(2):021003. Crossref, Google Scholar
• [19] Faulwasser T, Korda M, Jones CN, Bonvin D (2017) On turnpike and dissipativity properties of continuous-time optimal control problems. Automatica 81:297–304.Crossref, Google Scholar
• [20] Fletcher R (2010) The sequential quadratic programming method. Di Pillo G, Schoen F, eds. Nonlinear Optimization (Springer, Berlin, Heidelberg), 165–214.Crossref, Google Scholar
• [21] Geshkovski B, Zuazua E (2022) Turnpike in optimal control of PDEs, ResNets, and beyond. Acta Numer. 31:135–263.Crossref, Google Scholar
• [22] Gill PE, Wong E (2011) Sequential quadratic programming methods. Lee J, Leyffer S, eds. Mixed Integer Nonlinear Programming (Springer, New York), 147–224.Google Scholar
• [23] Grüne L (2013) Economic receding horizon control without terminal constraints. Automatica 49(3):725–734.Crossref, Google Scholar
• [24] Grüne L, Schaller M, Schiela A (2019) Sensitivity analysis of optimal control for a class of parabolic PDEs motivated by model predictive control. SIAM J. Control Optim. 57(4):2753–2774.Crossref, Google Scholar
• [25] Grüne L, Schaller M, Schiela A (2020) Exponential sensitivity and turnpike analysis for linear quadratic optimal control of general evolution equations. J. Differential Equations 268(12):7311–7341.Crossref, Google Scholar
• [26] Grüne L, Schaller M, Schiela A (2021) Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs. ESAIM COCV 27:56.Crossref, Google Scholar
• [27] Gunzburger MD (2002) Three approaches to optimal control and optimization. Perspectives in Flow Control and Optimization (Society for Industrial and Applied Mathematics, Philadelphia), 11–62.Crossref, Google Scholar
• [28] Hager WW (2000) Runge-Kutta methods in optimal control and the transformed adjoint system. Numer. Math. 87(2):247–282.Crossref, Google Scholar
• [29] Hernandez M (2018) Newton-Krylov PDE-constrained LDDMM in the space of band-limited vector fields. Preprint, submitted July 13, https://arxiv.org/abs/1807.05117.Google Scholar
• [30] Hernandez M (2019) A comparative study of different variants of Newton–Krylov PDE-constrained Stokes-LDDMM parameterized in the space of band-limited vector fields. SIAM J. Imaging Sci. 12(2):1038–1070.Crossref, Google Scholar
• [31] Herzog R, Kunisch K (2010) Algorithms for PDE‐constrained optimization. GAMM-Mitt. 33(2):163–176.Crossref, Google Scholar
• [32] Hintermüller M, Ito K, Kunisch K (2002) The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13(3):865–888.Crossref, Google Scholar
• [33] Hinze M, Kunisch K (2001) Second order methods for optimal control of time-dependent fluid flow. SIAM J. Control Optim. 40(3):925–946.Crossref, Google Scholar
• [34] Hinze M, Walther A, Sternberg J (2006) An optimal memory-reduced procedure for calculating adjoints of the instationary Navier-Stokes equations. Optim. Control Appl. Methods 27(1):19–40.Crossref, Google Scholar
• [35] Huerta I, Monzón P, Robledo G (2024) Controllability and feedback stabilizability in a nonuniform framework. Comptes Rendus. Mathématique 362(G12):1667–1692.Crossref, Google Scholar
• [36] Ito K, Kunisch K (2004) The primal-dual active set method for nonlinear optimal control problems with bilateral constraints. SIAM J. Control Optim. 43(1):357–376.Crossref, Google Scholar
• [37] Kirk DE (2004) Optimal Control Theory: An Introduction (Courier Corporation, Chelmsford, MA).Google Scholar
• [38] Lasdon L, Mitter S, Waren A (1967) The conjugate gradient method for optimal control problems. IEEE Trans. Automatic Control 12(2):132–138.Crossref, Google Scholar
• [39] LeVeque RJ (2007) The initial value problem for ordinary differential equations. Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (Society for Industrial and Applied Mathematics, Philadelphia), 113–135.Crossref, Google Scholar
• [40] Lui SH (1999) On Schwarz alternating methods for nonlinear elliptic PDEs. SIAM J. Sci. Comput. 21(4):1506–1523.Crossref, Google Scholar
• [41] Na S, Anitescu M (2020) Exponential decay in the sensitivity analysis of nonlinear dynamic programming. SIAM J. Optim. 30(2):1527–1554.Crossref, Google Scholar
• [42] Na S, Anitescu M, Kolar M (2024) A fast temporal decomposition procedure for long-horizon nonlinear dynamic programming. Math. Oper. Res. 49(2):1012–1044.Link, Google Scholar
• [43] Na S, Shin S, Anitescu M, Zavala VM (2022) On the convergence of overlapping Schwarz decomposition for nonlinear optimal control. IEEE Trans. Automatic Control 67(11):5996–6011.Crossref, Google Scholar
• [44] Neuberger JW (2009) Continuous steepest descent in Hilbert space: Nonlinear case. Sobolev Gradients and Differential Equations (Springer, Berlin, Heidelberg), 19–34.Google Scholar
• [45] Neuenhofen MP, Kerrigan EC (2020) An integral penalty-barrier direct transcription method for optimal control. 2020 59th IEEE Conf. Decision Control CDC (IEEE, Piscataway, NJ), 456–463.Google Scholar
• [46] Ni R, Na S, Shin S, Anitescu M (2024) Distributed sequential quadratic programming with overlapping graph decomposition and exact augmented Lagrangian. Preprint, submitted February 27, https://arxiv.org/abs/2402.17170v1.Google Scholar
• [47] Pazy A (1983) Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer, New York).Crossref, Google Scholar
• [48] Plessix RE (2006) A review of the adjoint-state method for computing the gradient of a functional with geophysical applications. Geophys. J. Internat. 167(2):495–503.Crossref, Google Scholar
• [49] Prudencio E, Byrd R, Cai XC (2005) Domain decomposition methods for PDE constrained optimization problems. Daydé M, Dongarra J, Hernández V, Palma JMLM, eds. High Performance Computing for Computational Science—VECPAR 2004 (Springer, Berlin, Heidelberg), 569–582.Crossref, Google Scholar
• [50] Sanz-Serna JM (2016) Symplectic Runge–Kutta schemes for adjoint equations, automatic differentiation, optimal control, and more. SIAM Rev. 58(1):3–33.Crossref, Google Scholar
• [51] Schorlepp T, Tong S, Grafke T, Stadler G (2023) Scalable methods for computing sharp extreme event probabilities in infinite-dimensional stochastic systems. Statist. Comput. 33(6):137. Crossref, Google Scholar
• [52] Shampine LF, Reichelt MW (1997) The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1):1–22.Crossref, Google Scholar
• [53] Shapiro S (1966) Lagrange and Mayer problems in optimal control. Automatica 3(3):219–230.Crossref, Google Scholar
• [54] Shen J, Tang T, Wang LL (2011) Spectral Methods: Algorithms, Analysis and Applications (Springer, Berlin, Heidelberg).Crossref, Google Scholar
• [55] Shin S, Anitescu M, Zavala VM (2020) Overlapping Schwarz decomposition for constrained quadratic programs. 2020 59th IEEE Conf. Decision Control CDC (IEEE, Piscataway, NJ), 3004–3009.Google Scholar
• [56] Shin S, Anitescu M, Zavala VM (2022) Exponential decay of sensitivity in graph-structured nonlinear programs. SIAM J. Optim. 32(2):1156–1183.Crossref, Google Scholar
• [57] Shin S, Faulwasser T, Zanon M, Zavala VM (2019) A parallel decomposition scheme for solving long-horizon optimal control problems. 2019 IEEE 58th Conf. Decision Control CDC (IEEE, Piscataway, NJ), 5264–5271.Google Scholar
• [58] Sontag ED (1998) Mathematical Control Theory (Springer, New York).Crossref, Google Scholar
• [59] Stakgold I, Holst M, eds. (2011) Green’s Functions and Boundary Value Problems (Wiley, Hoboken, NJ).Crossref, Google Scholar
• [60] Tröltzsch F (2010) Optimal Control of Partial Differential Equations (American Mathematical Society, Providence, RI).Crossref, Google Scholar
• [61] Ulbrich S (2009) Analytical background and optimality theory. Hinze M, Pinnau R, Ulbrich M, Ulbrich S, eds. Optimization with PDE Constraints (Springer Netherlands, Dordrecht, Netherlands), 1–95.Crossref, Google Scholar
• [62] Verschueren R, Zanon M, Quirynen R, Diehl M (2017) A sparsity preserving convexification procedure for indefinite quadratic programs arising in direct optimal control. SIAM J. Optim. 27(3):2085–2109.Crossref, Google Scholar
• [63] Wächter A, Biegler LT (2006) On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Programming 106(1):25–57. Crossref, Google Scholar
• [64] Xu W, Anitescu M (2018) Exponentially accurate temporal decomposition for long-horizon linear-quadratic dynamic optimization. SIAM J. Optim. 28(3):2541–2573.Crossref, Google Scholar
• [65] Zaitsev V (2015) Criteria for uniform complete controllability of a linear system. Vestnik Udmurtskogo Universiteta Matematika Mekhanika Komp’yuternye Nauki 25(2):157–179.Crossref, Google Scholar
• [66] Zeidler E (1990) Nonlinear Functional Analysis and Its Applications (Springer, New York).Google Scholar
• [67] Zhang F (2005) The Schur Complement and Its Applications (Springer-Verlag, New York).Crossref, Google Scholar