Abstract
Dynamic optimization problems are constrained by differential and algebraic equations and are found everywhere in science and engineering. A well-established method to solve these types of problems is direct transcription, where the differential equations are replaced with discretized approximations based on finite-difference or Runge–Kutta schemes. However, for problems with thousands of state variables and discretization points, direct transcription may result in nonlinear optimization problems which are too large for general-purpose optimization solvers to handle. Also, when an interior-point solver is applied, the dominant computational cost is solving the linear systems resulting from the Newton step. For large-scale nonlinear programming problems, these linear systems may become prohibitively expensive to solve. Furthermore, the systems also become too large to formulate and store in memory of a standard computer. On the other hand, direct transcription can exploit sparsity and structure of the linear systems in order to overcome these challenges. In this paper we investigate and compare two parallel linear decomposition algorithms, Cyclic Reduction (CR) and Schur complement decomposition, which take advantage of structure and sparsity. We describe the numerical conditioning of the CR algorithm when applied to the linear systems arising from dynamic optimization problems, and then compare CR with Schur complement decomposition on a number of test problems. Finally, we propose conditions under which each should be used, and describe future research directions.
Similar content being viewed by others
References
Amodio, P., Mazzia, F.: Backward error analysis of cyclic reduction for the solution of tridiagonal systems. Math. Comput. 62(206), 601–617 (1994)
Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: LAPACK Users’ Guide, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia (1999)
Betts, J.: Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, vol. 19. SIAM Series on Advances in Design and Control, Philadelphia (2010)
Biegler, L.T.: Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes. SIAM, Philadelphia (2010)
Biegler, L.T., Cervantes, A.M., Wächter, A.: Advances in simultaneous strategies for dynamic process optimization. Chem. Eng. Sci. 57(4), 575–593 (2002)
Binder, T., Blank, L., Bock, H.G., Bulirsch, R., Dahmen, W., Diehl, M., Kronseder, T., Marquardt, W., Schlöder, J.P., von Stryk, O.: Introduction to Model Based Optimization of Chemical Processes on Moving Horizons. In: Grötschel, M., Krumke, S.O., Rambau, J. (eds.) Online Optimization of Large Scale Systems, pp. 295–339. Springer, Berlin (2001)
Bini, D.A., Meini, B.: The cyclic reduction algorithm: from poisson equation to stochastic processes and beyond. Num. Algorithms 51(1), 23–60 (2009)
Borzi, A., Schulz, V.: Computational Optimization of Systems Governed by Partial Differential Equations. SIAM, Philadelphia (2012)
Bryson, A.E., Ho, Y.C.: Applied Optimal Control: Optimization, Estimation and Control. CRC Press, Boca Raton (1975)
Buneman, O.: Compact non-iterative Poisson solver. Technical Reports, Stanford University, California Institute for Plasma Research (1969)
Byrd, R.H., Hribar, M.E., Nocedal, J.: An interior point algorithm for large scale nonlinear programming. SIAM J. Opt. 9, 877–900 (1999)
Cannataro, B.Ş., Rao, A.V., Davis, T.A.: State-defect constraint pairing graph coarsening method for Karush–Kuhn–Tucker matrices arising in orthogonal collocation methods for optimal control. Comput. Optim. Appl. 64(3), 793–819 (2016)
Chang, S.C., Chang, T.S., Luh, P.B.: A hierarchical decomposition for large-scale optimal control problems with parallel processing structure. Automatica 25(1), 77–86 (1989)
Chiang, N., Petra, C.G., Zavala, V.M.: Structured nonconvex optimization of large-scale energy systems using PIPS-NLP. Power Syst. Comput. Conf. (PSCC) 2014, 1–7 (2014). https://doi.org/10.1109/PSCC.2014.7038374
Davis, T.A.: Algorithm 832: UMFPACK v4.3-an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30(2), 196–199 (2004). https://doi.org/10.1145/992200.992206
Diamond, M.A., Ferreira, D.L.V.: On a cyclic reduction method for the solution of Poisson’s equations. SIAM J. Numer. Anal. 13(1), 54–70 (1976)
Diehl, M., Bock, H.G., Schlöder, J.P., Findeisen, R., Nagy, Z., Allgöwer, F.: Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations. J. Process Control 12(4), 577–585 (2002)
Duff, I.S.: MA57–a code for the solution of sparse symmetric definite and indefinite systems. ACM Trans. Math. Softw. 30(2), 118–144 (2004). https://doi.org/10.1145/992200.992202
Feng, D., Schnabel, R.B.: Globally convergent parallel algorithms for solving block bordered systems of nonlinear equations. Optim. Methods Softw. 2(3–4), 269–295 (1993)
Fiacco, A., McCormick, G.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York (1968)
Frasch, J.: Parallel Algorithms for Optimization of Dynamic Systems in Real-Time. Ph.D. thesis, Otto-von-Guericke University Magdeburg (2014). http://www.mathopt.de/PUBLICATIONS/Frasch2014.pdf
Gondzio, J.: Matrix-free interior point method. Comput. Optim. Appl. 51(2), 457–480 (2012)
Hager, W.W., Hou, H., Rao, A.V.: Lebesgue constants arising in a class of collocation methods. IMA J. Numer. Anal. 37, 1884 (2016)
Hartwich, A., Stockmann, K., Terboven, C., Feuerriegel, S., Marquardt, W.: Parallel sensitivity analysis for efficient large-scale dynamic optimization. Optim. Eng. 12(4), 489–508 (2011)
Heller, D.: Some aspects of the cyclic reduction algorithm for block tridiagonal linear systems. SIAM J. Numer. Anal. 13(4), 484–496 (1976)
Hirshman, S.P., Perumalla, K.S., Lynch, V.E., Sánchez, R.: Bcyclic: a parallel block tridiagonal matrix cyclic solver. J. Comput. Phys. 229(18), 6392–6404 (2010)
Hockney, R., Jesshope, C.: Parallel computers: architecture, programming and algorithms. Adam Hilger (1981). https://books.google.com/books?id=CP5gzotAXI4C
Hockney, R.W.: A fast direct solution of poisson’s equation using fourier analysis. J. ACM 12(1), 95–113 (1965). https://doi.org/10.1145/321250.321259
Kang, J., Cao, Y., Word, D.P., Laird, C.D.: An interior-point method for efficient solution of block-structured nlp problems using an implicit schur-complement decomposition. Comput. Chem. Eng. 71, 563–573 (2014)
Kang, J., Chiang, N., Laird, C.D., Zavala, V.M.: Nonlinear programming strategies on high-performance computers. In: IEEE 54th annual conference on decision and control (CDC), pp. 4612–4620 (2015)
Körkel, S., Kostina, E., Bock, H., Schlöder, J.: Numerical methods for optimal control problems in design of robust optimal experiments for nonlinear dynamic processes. Optim. Methods Softw. 19(34), 327338 (2004)
Laird, C.D., Wong, A.V., Åkesson, J.: Parallel solution of large-scale dynamic optimization problems. In: 21st European symposium on computer-aided process engineering (2011)
Le, D.D.: Parallelisierung von Innere–Punkte–Verfahren mittels Cyclic Reduction. Bachelor’s thesis, OvGU Magdeburg (2014). http://www.mathopt.de/PUBLICATIONS/Do2014.pdf
Leineweber, D.B., Bauer, I., Bock, H.G., Schlöder, J.P.: An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part 1: theoretical aspects. Comput. Chem. Eng. 27(2), 157–166 (2003)
Leineweber, D.B., Schäfer, A., Bock, H.G., Schlöder, J.P.: An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization: Part II: software aspects and applications. Comput. Chem. Eng. 27(2), 167–174 (2003)
Leugering, G., Engell, S., Griewank, A., Hinze, M., Rannacher, R., Schulz, V., Ulbrich, M., Ulbrich, S.: Constrained Optimization and Optimal Control for Partial Differential Equations. Springer, Basel (2016). https://doi.org/10.1007/978-3-0348-0133-1
MATLAB: version 8.3.0 (R2012a). The MathWorks Inc., Natick, Massachusetts (2012)
MATLAB: MATLAB function reference, pp. 5702–5709. The MathWorks Incorporated (2015)
Pearson, J.W., Gondzio, J.: Fast interior point solution of quadratic programming problems arising from pde-constrained optimization. Numerische Mathematik pp. 1–41 (2016)
Petra, C.G., Schenk, O., Anitescu, M.: Real-time stochastic optimization of complex energy systems on high-performance computers. Comput. Sci. Eng. 16(5), 32–42 (2014)
Pontryagin, L., Boltyanskii, V., Gamkrelidze, R., Mishchenko, E.: Mathematical Theory of Optimal Processes. Interscience Publishers Inc., New York (1962)
Qin, S.J., Badgwell, T.A.: An Overview of Nonlinear Model Predictive Control Applications. In: Allgöwer, F., Zheng, A. (eds.) Nonlinear Model Predictive Control, pp. 369–392. Springer, Berlin (2000)
Rosmond, T.E., Faulkner, F.D.: Direct solution of elliptic equations by block cyclic reduction and factorization. Mon. Weather Rev. 104(5), 641–649 (1976)
Steinbach, M.C.: A structured interior point SQP method for nonlinear optimal control problems. In: Computational Optimal Control, vol. 115 of International Series of Numerical Mathematics, pp. 213–222. Birkhäuser (1994)
Steinbach, M.C.: Structured interior point sqp methods in optimal control. Z. Angew. Math. Mech 76(S3), 59–62 (1996)
Swarztrauber, P.N.: A direct method for the discrete solution of separable elliptic equations. SIAM J. Numer. Anal. 11(6), 1136–1150 (1974)
Swarztrauber, P.N., Sweet, R.A.: Algorithm 541: efficient fortran subprograms for the solution of separable elliptic partial differential equations [d3]. ACM Trans. Math. Softw. 5(3), 352–364 (1979). https://doi.org/10.1145/355841.355850
Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106, 25–57 (2006)
Word, D.P., Kang, J., Akesson, J., Laird, C.D.: Efficient parallel solution of large-scale nonlinear dynamic optimization problems. Comput. Optim. Appl. 59(3), 667–688 (2014)
Wright, S.J.: Partitioned dynamic programming for optimal control. SIAM J. Optim. 1(4), 620–642 (1991)
Yalamov, P., Pavlov, V.: Stability of the block cyclic reduction. Linear Algebra Appl. 249(1), 341–358 (1996)
Zavala, V.M., Laird, C.D., Biegler, L.T.: Interior-point decomposition approaches for parallel solution of large-scale nonlinear parameter estimation problems. Chem. Eng. Sci. 63(19), 4834–4845 (2008)
Zhang, X., Byrd, R.H., Schnabel, R.B.: Parallel methods for solving nonlinear block bordered systems of equations. SIAM J. Sci. Stat. Comput. 13(4), 841–859 (1992)
Zhang, Y., Cohen, J., Owens, J.D.: Fast tridiagonal solvers on the GPU. ACM Sigplan Notices 45(5), 127–136 (2010)
Acknowledgements
Funding from ExxonMobil Research and Engineering Company is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nicholson, B.L., Wan, W., Kameswaran, S. et al. Parallel cyclic reduction strategies for linear systems that arise in dynamic optimization problems. Comput Optim Appl 70, 321–350 (2018). https://doi.org/10.1007/s10589-018-0001-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-018-0001-7