Mathematical Programming Computation

, Volume 3, Issue 4, pp 319–348 | Cite as

A factorization with update procedures for a KKT matrix arising in direct optimal control

  • Christian Kirches
  • Hans Georg Bock
  • Johannes P. Schlöder
  • Sebastian Sager
Full Length Paper

Abstract

Quadratic programs obtained for optimal control problems of dynamic or discrete-time processes usually involve highly block structured Hessian and constraints matrices, to be exploited by efficient numerical methods. In interior point methods, this is elegantly achieved by the widespread availability of advanced sparse symmetric indefinite factorization codes. For active set methods, however, conventional dense matrix techniques suffer from the need to update base matrices in every active set iteration, thereby loosing the sparsity structure after a few updates. This contribution presents a new factorization of a KKT matrix arising in active set methods for optimal control. It fully respects the block structure without any fill-in. For this factorization, matrix updates are derived for all cases of active set changes. This allows for the design of a highly efficient block structured active set method for optimal control and model predictive control problems with long horizons or many control parameters.

Keywords

Matrix factorizations and updates Block structured active set quadratic programming Direct methods for optimal control 

Mathematics Subject Classification (2010)

90C20 90C30 93B40 15A23 65F05 

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References

  1. 1.
    Albersmeyer, J., Bock, H.: Efficient sensitivity generation for large scale dynamic systems. Technical report, SPP 1253 Preprints, University of Erlangen (2009)Google Scholar
  2. 2.
    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)Google Scholar
  3. 3.
    Bartlett R., Biegler L.: QPSchur: a dual, active set, Schur complement method for large-scale and structured convex quadratic programming algorithm. Optim. Eng. 7, 5–32 (2006)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bartlett, R., Wächter, A., Biegler, L.: Active set vs. interior point strategies for model predictive control. In: Proceedings of the American Control Conference, Chicago, IL, pp. 4229–4233 (2000)Google Scholar
  5. 5.
    Benzi M., Golub G., Liesen J.: Numerical solution of saddle-point problems. Acta Numerica 14, 1–137 (2005)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Best M.: An algorithm for the solution of the parametric quadratic programming problem. In: Fischer, H., Riedmüller, B., Schäffler, S. (eds.) Applied Mathematics and Parallel Computing—Festschrift for Klaus Ritter, Chap. 3, pp. 57–76. Physica-Verlag, Heidelberg (1996)Google Scholar
  7. 7.
    Bock, H., Plitt, K.: A multiple shooting algorithm for direct solution of optimal control problems. In: Proceedings of the 9th IFAC World Congress, pp. 242–247. Pergamon Press, Budapest (1984)Google Scholar
  8. 8.
    Bock H., Diehl M., Kostina E., Schlöder J.: Constrained optimal feedback control for DAE. In: Biegler, L., Ghattas, O., Heinkenschloss, M., Keyes, D., van Bloemen Waanders, B. (eds.) Real-Time PDE-Constrained Optimization, Chap. 1, pp. 3–24. SIAM, Philadelphia (2007)CrossRefGoogle Scholar
  9. 9.
    Davis T.: Algorithm 832: UMFPACK—an unsymmetric-pattern multifrontal method with a column pre-ordering strategy. ACM Trans. Math. Softw. 30, 196–199 (2004)MATHCrossRefGoogle Scholar
  10. 10.
    Davis T., Hager W.: Multiple-rank modifications of a sparse Cholesky factorization. SIAM J. Matrix Anal. Appl. 22(4), 997–1013 (2000)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Diehl M., Bock H., Schlöder J., Findeisen R., Nagy Z., Allgöwer F.: Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations. J. Proc. Contr. 12(4), 577–585 (2002)CrossRefGoogle Scholar
  12. 12.
    Diehl M., Kuehl P., Bock H., Schlöder J.: Schnelle Algorithmen für die Zustands- und Parameterschätzung auf bewegten Horizonten. Automatisierungstechnik 54(12), 602–613 (2006)CrossRefGoogle Scholar
  13. 13.
    Duff I.: MA57—a code for the solution of sparse symmetric definite and indefinite systems. ACM Trans. Math. Softw. 30(2), 118–144 (2004)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Duff I., Reid J.: The multifrontal solution of indefinite sparse symmetric linear equations. ACM Trans. Math. Softw. 9(3), 302–325 (1983)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Eldersveld S., Saunders M.: A block-LU update for large scale linear programming. SIAM J. Matrix Anal. Appl. 13, 191–201 (1992)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Ferreau H., Bock H., Diehl M.: An online active set strategy to overcome the limitations of explicit MPC. Int. J. Robust Nonlinear Control 18(8), 816–830 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Fletcher R.: Practical Methods of Optimization, 2nd edn. Wiley, Chichester (1987)MATHGoogle Scholar
  18. 18.
    Fletcher, R.: Resolving degeneracy in quadratic programming. Numerical Analysis Report NA/135, University of Dundee, Dundee, Scotland (1991)Google Scholar
  19. 19.
    Fletcher, R.: Approximation Theory and Optimization. Dense Factors of Sparse Matrices, pp. 145–166. Tributes to M.J.D. Powell. Cambridge University Press (1997)Google Scholar
  20. 20.
    Fletcher, R.: Numerical Analysis 1997. Block Triangular Orderings and Factors for Sparse Matrices in LP, pp. 91–110. Pitman Research Notes in Mathematics, vol. 380. Longman, Harlow (1998)Google Scholar
  21. 21.
    Gerdts M.: Solving mixed-integer optimal control problems by Branch&Bound: a case study from automobile test-driving with gear shift. Optimal Control Appl. Methods 26, 1–18 (2005)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gertz E., Wright S.: Object-oriented software for quadratic programming. ACM Trans. Math. Softw. 29, 58–81 (2003)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Gill P., Golub G., Murray W., Saunders M.A.: Methods for modifying matrix factorizations. Math. Comput. 28(126), 505–535 (1974)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Gill P., Murray W., Saunders M., Wright M.: Sparse matrix methods in optimization. SIAM J. Sci. Stat. Comput. 5(3), 562–589 (1984)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Gill P., Murray W., Saunders M., Wright M.: A practical anti-cycling procedure for linearly constrained optimization. Math. Program. 45(1–3), 437–474 (1989)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Gill P., Murray W., Saunders M., Wright M.: Inertia-controlling methods for general quadratic programming. SIAM Rev. 33(1), 1–36 (1991)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Gill, P., Murray, W., Saunders, M.: User’s Guide For QPOPT 1.0: A Fortran Package for Quadratic Programming (1995)Google Scholar
  28. 28.
    Golub G., van Loan C.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)MATHGoogle Scholar
  29. 29.
    Hall J., McKinnon K.: The simplest examples where the simplex method cycles and conditions where the EXPAND method fails to prevent cycling. Math. Program. Ser. A & B 100(1), 133–150 (2004)MathSciNetMATHGoogle Scholar
  30. 30.
    Han S.: Superlinearly convergent variable-metric algorithms for general nonlinear programming problems. Math. Program. 11, 263–282 (1976)CrossRefGoogle Scholar
  31. 31.
    Haseltine E., Rawlings J.: Critical evaluation of extended Kalman filtering and moving-horizon estimation. Ind. Eng. Chem. Res. 44, 2451–2460 (2005)CrossRefGoogle Scholar
  32. 32.
    Huynh, H.: A large-scale quadratic programming solver based on block-LU updates of the KKT system. PhD thesis, Stanford University (2008)Google Scholar
  33. 33.
    Kirches C., Sager S., Bock H., Schlöder J.: Time-optimal control of automobile test drives with gear shifts. Optimal Control Appl. Methods 31(2), 137–153 (2010)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Kirches C., Bock H., Schlöder J., Sager S.: Block structured quadratic programming for the direct multiple shooting method for optimal control. Optim. Methods Softw. 26(2), 239–257 (2011)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Leineweber D., Bauer I., Schäfer A., Bock H., Schlöder J.: An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization (Parts I and II). Comput. Chem. Eng. 27, 157–174 (2003)CrossRefGoogle Scholar
  36. 36.
    Nocedal J., Wright S.: Numerical Optimization, 2nd edn. Springer, Berlin, Heidelberg, New York (2006)MATHGoogle Scholar
  37. 37.
    Powell M.: Algorithms for nonlinear constraints that use Lagrangian functions. Math. Program. 14(3), 224–248 (1978)MATHCrossRefGoogle Scholar
  38. 38.
    Powell, M.: ZQPCVX: a Fortran subroutine for convex quadratic programming. Technical report, Department of Applied Mathematics and Theoretical Physics, Cambridge University (1983)Google Scholar
  39. 39.
    Schmid C., Biegler L.: Quadratic programming methods for tailored reduced Hessian SQP. Comput. Chem. Eng. 18(9), 817–832 (1994)CrossRefGoogle Scholar
  40. 40.
    Steinbach, M.: Fast recursive SQP methods for large-scale optimal control problems. PhD thesis, Ruprecht-Karls-Universität Heidelberg (1995)Google Scholar
  41. 41.
    Vanderbei R.: LOQO: an interior point code for quadratic programming. Optim. Methods Softw. 11(1–4), 451–484 (1999)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Wächter A., Biegler L.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Wirsching, L., Albersmeyer, J., Kühl, P., Diehl, M., Bock, H.: An adjoint-based numerical method for fast nonlinear model predictive control. In: Chung, M., Misra, P. (eds.) Proceedings of the 17th IFAC World Congress, Seoul, Korea, July 6–11, 2008. IFAC-PapersOnLine, vol. 17, pp. 1934–1939 (2008)Google Scholar
  44. 44.
    Wright, S.: Applying new optimization algorithms to model predictive control. In: Fifth International Conference on Chemical Process Control—CPC V, pp. 147–155. CACHE Publications (1997)Google Scholar

Copyright information

© Springer and Mathematical Optimization Society 2011

Authors and Affiliations

  • Christian Kirches
    • 1
  • Hans Georg Bock
    • 1
  • Johannes P. Schlöder
    • 1
  • Sebastian Sager
    • 1
  1. 1.Interdisciplinary Center for Scientific Computing (IWR)Heidelberg UniversityHeidelbergGermany

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