Abstract
This paper describes the design of a “sparse” nonlinear programming algorithm for a “large scale” flight optimization and sizing code. This program is designed to solve trajectory optimal control problems with path constraints. The direct transcription technique is applied to discretize an optimal control problem into a sparse, large scale parameter optimization problem for subsequent numerical solution by a nonlinear programming algorithm. This paper focuses on the design of a generalized reduced gradient algorithm to exploit the sparsity structure of the collocation equations.
This work was supported in part by an Aerospace Sponsored Research Grant.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Abadie and J. Carpentier; Generalization of the Wolfe reduced-gradient method to the case of nonlinear constraints, Optimization (Ed. by R. Fletcher), Academic, 1969.
U. Ascher, R. Mattheij, and R. Russell; Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice Hall, Engle-wood Cliffs, 1988.
T. Beltracchi; A comparison of the generalized reduced gradient and generalized projected gradient methods, to appear.
T. Beltracchi; Optimization software modifications to study launch vehicle sizing trajectory design problems, AIAA Paper 94–4403, 5th AIAA /NASA/ USAF/ISSMO Symposium on Multidisciplinary Analysis and Optimization, 1994.
J. Betts and P. Frank; A sparse nonlinear optimization algorithm, AMS-TR-173, Boeing Computer Services, 1991.
J. Betts and W. Hallman; NLP2 optimization algorithm documentation, The Aerospace Corporation, Report No. TOR-0089(4464-06)-1, 1989.
J. Betts and W. Huffman; The application of sparse nonlinear programming to trajectory optimization, J. Guid. Cont. Dynam. 15, 1992.
K. Brenan; Differential-algebraic equations issues in the direct transcription of path constrained optimal control problems, Ann. Num. Math. 1, 1994, 247–263.
K. Brenan, S. Campbell, and L. Petzold; Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Elsevier, New York, 1989.
K. Brenan and W. Hallman; A large-scale generalized reduced gradient algorithm for flight optimization and sizing problems, The Aerospace Corporation, Report No. ATR-94(8489)-3, 1994.
K. Brenan, W. Hallman, and W. Yeung; The design of a large-scale NLP code for trajectory optimization problems, The Aerospace Corporation, Report No. ATR-92(8189)-1, 1992.
A. Curtis, M. Powell, and J. Reid; On the estimation of sparse Jacobian matrices, J. Inst. Math. Appl. 13, 1974, 117–119.
C. De Boor and R. Weiss; SOLVEBLOK: a package for solving almost block diagonal linear systems, ACM Trans. Math. Soft. 6, 1980, 80–87.
W. Govaerts and J. Pryce; Mixed block elimination for linear systems with wider borders, IMA J. Num. Anal. 13, 1993, 161–180.
W. Hallman; Optimization algorithm NLP3, The Aerospace Corporation, IOC A87-5752-45, 1987. (Aerospace internal document; not available for external distribution.)
W. Hallman; Optimal scaling techniques for the nonlinear programming problem, AIAA Paper 94-4417, 5th AIAA /NASA /USAF/ISSMO Symposium on Multidisciplinary Analysis and Optimization, 1994.
C. Hargraves and S. Paris; OTTS-optimal trajectories by implicit integration, Boeing Aerospace Co., Contract No. F33615-85-c-3009, 1988.
C. Hargraves and S. Paris; Direct trajectory optimization using nonlinear programming and collocation, AIAA J. Guid. Cont. Dynam. 10, 1987, 338–342.
N. Higham; A survey of condition number estimation for triangular matrices, SIAM Review 29, 1987.
C. Lawson and R. Hanson; Solving Least Squares Problems, Prentice-Hall, Englewood Cliffs, 1974.
B. Murtagh and M. Saunders; MINOS/AUGMENTED user’s manual, Report SOL 80-14, Department of Operations Research, Stanford, 1980.
H. Nguyen; FONSIZE: a trajectory optimization and vehicle sizing program, The Aerospace Corporation, Report No. TOR-0091(6911-04)-1, 1991.
Y. Shi, R. Nelson, D. Young, P. Gill, W. Murray, and M. Saunders; The application of nonlinear programming and collocation to optimal aeroassisted orbital transfer trajectory, AIAA Paper 92-0734, 1992.
J. Tomlin; Pivoting for sparsity and size in linear programming inversion routines, J. Inst. Maths. Applics. 10, 1972, 289–295.
J. Tomlin; Maintaining a sparse inverse in the simplex method, IBM J. Res. Dev. 16, 1972,415–423.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media New York
About this paper
Cite this paper
Brenan, K.E., Hallman, W.P. (1995). A Generalized Reduced Gradient Algorithm for Large-scale Trajectory Optimization Problems. In: Borggaard, J., Burkardt, J., Gunzburger, M., Peterson, J. (eds) Optimal Design and Control. Progress in Systems and Control Theory, vol 19. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0839-6_7
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0839-6_7
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6916-8
Online ISBN: 978-1-4612-0839-6
eBook Packages: Springer Book Archive