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

Mathematical Programming Computation

Volumn 3, Issue 4, 2011, Pages 319-348

  Kirches, Christian ^a   Bock, Hans Georg ^a   Schlöder, Johannes P ^a   Sager, Sebastian ^a  

^a UNIVERSITY OF HEIDELBERG   (Germany)

Author keywords

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

Indexed keywords

FACTORIZATION; ITERATIVE METHODS; MODEL PREDICTIVE CONTROL; NUMERICAL METHODS; OPTIMAL CONTROL SYSTEMS; QUADRATIC PROGRAMMING;

ACTIVE SETS; EFFICIENT NUMERICAL METHOD; INTERIOR-POINT METHOD; MATRIX FACTORIZATIONS; OPTIMAL CONTROL PROBLEM; OPTIMAL CONTROLS; QUADRATIC PROGRAMS; SYMMETRIC INDEFINITE FACTORIZATIONS;

MATRIX ALGEBRA;

EID: 84857792231     PISSN: 18672949     EISSN: 18672957     Source Type: Journal    
DOI: 10.1007/s12532-011-0030-z     Document Type: Article

References (44)

1. Albersmeyer, J., Bock, H.: Efficient sensitivity generation for large scale dynamic systems. Technical report, SPP 1253 Preprints, University of Erlangen (2009)
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)
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)
4. Bartlett, R., Wächter, A., Biegler, L.: Active set vs. interior point strategies formodel predictive control. In: Proceedings of the American Control Conference, Chicago, IL, pp. 4229-4233 (2000)
5. Benzi, M., Golub, G., Liesen, J.: Numerical solution of saddle-point problems. Acta Numerica 14, 1-137 (2005)
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)
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)
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)
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)
10. Davis, T., Hager, W.: Multiple-rank modifications of a sparse Cholesky factorization. SIAM J. Matrix Anal. Appl. 22(4), 997-1013 (2000)
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)
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)
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)
14. Duff, I., Reid, J.: Themultifrontal solution of indefinite sparse symmetric linear equations. ACMTrans. Math. Softw. 9(3), 302-325 (1983)
15. Eldersveld, S., Saunders, M.: A block-LU update for large scale linear programming. SIAM J. Matrix Anal. Appl. 13, 191-201 (1992)
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)
17. Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley, Chichester (1987)
18. Fletcher, R.: Resolving degeneracy in quadratic programming. Numerical Analysis Report NA/135, University of Dundee, Dundee, Scotland (1991)
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)
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)
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)
22. Gertz, E., Wright, S.: Object-oriented software for quadratic programming. ACM Trans. Math. Softw. 29, 58-81 (2003)
23. Gill, P., Golub, G., Murray, W., Saunders, M.A.: Methods for modifying matrix factorizations. Math. Comput. 28(126), 505-535 (1974)
24. Gill, P., Murray, W., Saunders, M., Wright, M.: Sparse matrix methods in optimization. SIAM J. Sci. Stat. Comput. 5(3), 562-589 (1984)
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)
26. Gill, P., Murray, W., Saunders, M., Wright, M.: Inertia-controlling methods for general quadratic programming. SIAM Rev. 33(1), 1-36 (1991)
27. Gill, P., Murray, W., Saunders, M.: User's Guide For QPOPT 1.0: A Fortran Package for Quadratic Programming (1995)
28. Golub, G., van Loan, C.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)
29. Hall, J., McKinnon, K.: The simplest examples where the simplexmethod cycles and conditions where the EXPAND method fails to prevent cycling. Math. Program. Ser. A & B 100(1), 133-150 (2004)
30. Han, S.: Superlinearly convergent variable-metric algorithms for general nonlinear programming problems. Math. Program. 11, 263-282 (1976)
31. Haseltine, E., Rawlings, J.: Critical evaluation of extended Kalman filtering and moving-horizon estimation. Ind. Eng. Chem. Res. 44, 2451-2460 (2005)
32. Huynh, H.:Alarge-scale quadratic programming solver based on block-LU updates of theKKTsystem. PhD thesis, Stanford University (2008)
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)
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)
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)
36. Nocedal, J., Wright, S.: Numerical Optimization, 2nd edn. Springer, Berlin, Heidelberg, New York (2006)
37. Powell, M.: Algorithms for nonlinear constraints that use Lagrangian functions. Math. Program. 14(3), 224-248 (1978)
38. Powell, M.: ZQPCVX: a Fortran subroutine for convex quadratic programming. Technical report, Department of Applied Mathematics and Theoretical Physics, Cambridge University (1983)
39. Schmid, C., Biegler, L.: Quadratic programming methods for tailored reduced Hessian SQP. Comput. Chem. Eng. 18(9), 817-832 (1994)
40. Steinbach, M.: Fast recursive SQP methods for large-scale optimal control problems. PhD thesis, Ruprecht-Karls-Universität Heidelberg (1995)
41. Vanderbei, R.: LOQO: an interior point code for quadratic programming. Optim. Methods Softw. 11(1-4), 451-484 (1999)
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)
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)
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)