Inexact central path following algorithms for optimal control problems

SIAM Journal on Control and Optimization

Volumn 46, Issue 3, 2007, Pages 792-815

  Weiser, Martin ^a   Deuflhard, Peter ^b  

^a ZUSE INSTITUTE BERLIN   (Germany)

^b FREIE UNIVERSITÄT BERLIN   (Germany)

Author keywords

Affine invariant Newton methods; Interior point methods in function space; Numerical optimal control

Indexed keywords

ALGORITHMS; EVOLUTIONARY ALGORITHMS; FUNCTION EVALUATION; HYBRID SENSORS; INTERSECTIONS; NEWTON-RAPHSON METHOD; NUMBER THEORY; TREES (MATHEMATICS);

(E ,2E) THEORY; ADAPTIVE STEP SIZE; AFFINE INVARIANTS; APPLIED (CO); APPLIED MATHEMATICS; CENTRAL PATH; DIRECT AND INDIRECT METHODS; DUALITY GAPS; FUNCTION SPACES; HYBRID METHODS; INEXACT NEWTON METHODS; INTERIOR POINT; NEW APPROACHES; NUMERICAL CONTINUATION; NUMERICAL SOLUTIONS; OPTIMAL CONTROL PROBLEM (OCP); STATE CONSTRAINTS; WIND SHEAR;

NUMERICAL METHODS;

EID: 44649087708     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012903396851     Document Type: Article

References (42)

1. W. ALT AND K. MALANOWSKI, The Lagrange-Newton method for state constrained optimal control problems, Comput. Optim. Appl., 4 (1995), pp. 217-239.
2. U. ASCHER, J. CHRISTIANSEN, AND R. RUSSELL, Collocation software for boundary-value ODEs, ACM Trans. Math. Softw., 7 (1981), pp. 209-222.
3. U. ASCHER, R. MATTHEIJ, AND R. RUSSELL, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1988.
4. G. BADER AND U. ASCHER, A new basis implementation for a mixed order boundary value ODE solver, SIAM J. Sci. Statist. Comput., 8 (1987), pp. 483-500.
5. M. BERGOUNIOUX, M. HADDOU, M. HINTERMÜ LLER, AND K. KUNISCH, A comparison of a Moreau-Yosida-based active set strategy and interior point methods for constrained optimal control problems, SIAM J. Optim., 11 (2000), pp. 495-521.
6. P. BERKMANN AND H. PESCH, Abort landing in windshear: An optimal control problem with third-order state constraint and varied switching structure, J. Optim. Theory Appl., 85 (1995), pp. 21-57.
7. H. BOCK, Numerische Behandlung von zustandsbeschränkten und Chebychef-Steuerungs-Problemen, tech. rep., Carl-Cranz-Gesellschaft, Oberpfaffenhofen, Germany, 1981.
8. H. BOCK AND K.-J. PLITT, A multiple shooting algorithm for direct solution of optimal control problems, in Proceedings of the 9th IFAC World Congress, Budapest, Pergamon Press, Oxford, 1984, pp. 242-247.
9. R. BULIRSCH, Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertprob-lernen und Aufgaben der optimalen Steuerung, tech. rep., Carl-Cranz-Gesellschaft, Oberpfaffenhofen, Germany, 1971.
10. R. BULIRSCH, F. MONTRONE, AND H. PESCH, Abort landing in the presence of windshear as a minimax optimal control problem. I: Necessary conditions, J. Optim. Theory Appl., 70 (1991), pp. 1-23.
11. R. BULIRSCH, F. MONTRONE, AND H. PESCH, Abort landing in the presence of windshear as a minimax optimal control problem. II: Multiple shooting and homotopy, J. Optim. Theory Appl., 70 (1991), pp. 223-254.
12. R. BULIRSCH, E. NERZ, H. PESCH, AND O. VON STRYK, Combining direct and indirect methods in optimal control: Range maximization of a hang glider, in Optimal Control, Internat. Ser. Numer. Math. 111, R. Bulirsch, A. Miele, J. Stoer, and K. H. Well, eds., Birkhäuser, Basel, 1993, pp. 273-288.
13. J. BURKE AND S. XU, The global linear convergence of a noninterior path-following algorithm for linear complementarity problems, Math. Oper. Res., 23 (1998), pp. 719-734.
14. B. CHEN AND N. XIU, A global linear and local quadratic noninterior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions, SIAM J. Optim., 9 (1999), pp. 605-623.
15. C. CHEN AND O. L. MANGASARIAN, A class of smoothing functions for nonlinear and mixed complementarity problems, Comput. Optim. Appl., 5 (1996), pp. 97-138.
16. P. DEUFLHARD, A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting, Numer. Math., 22 (1974), pp. 289-315.
17. P. DEUFLHARD, A stepsize control for continuation methods and its special application to multiple shooting techniques, Numer. Math., 33 (1979), pp. 115-146.
18. P. DEUFLHARD, Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms, Comput. Math. 35, Springer, Berlin, 2004.
19. P. DEUFLHARD AND G. HEINDL, Affine invariant convergence theorems for Newton's method and extensions to related methods, SIAM J. Numer. Anal., 16 (1979), pp. 1-10.
20. P. DEUFLHARD AND M. WEISER, Local inexact Newton multilevel FEM for nonlinear elliptic problems, in Computational Science for the 21st Century, M.-O. Bristeau, G. Etgen, W. Fitzgibbon, J.-L. Lions, J. Periaux, and M. Wheeler, eds., Wiley, Chichester, UK, 1997, pp. 129-138.
21. A. FISCHER, A special Newton-type optimization method, Optimization, 24 (1992), pp. 269-284.
22. W. HAGER, Runge-Kutta methods in optimal control and the transformed adjoint system, Numer. Math., 87 (2000), pp. 247-282.
23. W. HAGER, Numerical analysis in optimal control, in Proceedings of the Conference on Optimal Control of Complex Structures, Internat. Ser. Numer. Math. 139, K.-H. Hoffmann, I. Lasiecka, G. Leugering, J. Sprekels, and F. Tröltzsch, eds., Birkhäuser, Basel, 2001, pp. 83-93.
24. A. HOHMANN, Inexact Gauss Newton Methods for Parameter Dependent Nonlinear Problems, Ph.D. thesis, Free University of Berlin, Berlin, Germany, 1994.
25. S. ITO, C. KELLEY, AND E. SACHS, Inexact primal-dual interior point iteration for linear programs in function spaces, Comput. Optim. Appl., 4 (1995), pp. 189-201.
26. F. JARRE, Comparing Two Interior-Point Approaches for Semi-infinite Programs, tech. rep., Universität Trier, Trier, Germany, 1999.
27. C. KANZOW, Some noninterior continuation methods for linear complementarity problems, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 851-868.
28. S. KURCYUSZ, On the existence and nonexistence of Lagrange multipliers in Banach spaces, J. Optim. Theory Appl., 20 (1976), pp. 81-110.
29. H. MAURER, First and second order sufficient optimality conditions in mathematical programming and optimal control, Math. Programming Stud., 14 (1981), pp. 163-177.
30. A. MIELE, T. WANG, C. TZENG, AND W. MELVIN, Optimal abort landing trajectories in the presence of a windshear, J. Optim. Theory Appl., 55 (1987), pp. 165-202.
31. F. POTRA, A Path-Following Method for Linear Complementarity Problems Based on the Affine Invariant Kantorovich Theorem, ZIB Report 00-30, Zuse Institute Berlin, Berlin, Germany, 2000.
32. R. PYTLAK, Numerical Methods for Optimal Control Problems with State Constraints, Lecture Notes in Math. 1707, Springer, New York, 1999.
33. V. SCHULZ, Solving discretized optimization problems by partially reduced SQP methods, Comput. Vis. Sci., 1 (1998), pp. 83-96.
34. J. STOER AND R. BULIRSCH, Introduction to Numerical Analysis, Springer, New York, 1993.
35. M. ULBRICH, Semismooth Newton methods for operator equations in function spaces, SIAM J. Optim., 13 (2003), pp. 805-842.
36. M. ULBRICH AND S. ULBRICH, Superlinear convergence of affine-scaling interior-point Newton methods for infinite-dimensional nonlinear problems with pointwise bounds, SIAM J. Control Optim., 38 (2000), pp. 1938-1984.
37. M. ULBRICH, S. ULBRICH, AND M. HEINKENSCHLOSS, Global convergence of trust-region interior-point algorithms for infinite-dimensional nonconvex minimization subject to point-wise bounds, SIAM J. Control Optim., 37 (1999), pp. 731-764.
38. S. VOLKWEIN AND M. WEISER, Affine invariant convergence analysis for inexact augmented Lagrangian-SQP methods, SIAM J. Control Optim., 41 (2002), pp. 875-899.
39. O. VON STRYK AND R. BULIRSCH, Direct and indirect methods for trajectory optimization, Ann. Oper. Res., 37 (1992), pp. 357-373.
40. M. WEISER, Function Space Complementarity Methods for Optimal Control Problems, doctoral thesis, Free University of Berlin, Berlin, Germany, 2001.
41. M. WEISER, Linear Convergence of an Interior Point Method for Linear Control Constrained Optimal Control Problems, TAB Report 02-13, Zuse Institute Berlin, Berlin, Germany, 2002.
42. M. WEISER, Interior point methods in function space, SIAM J. Control Optim., 44 (2005), pp. 1766-1786.