A primal-dual active set algorithm for bilaterally control constrained optimal control problems
Author:
Michael Hintermüller
Journal:
Quart. Appl. Math. 61 (2003), 131-160
MSC:
Primary 49M37; Secondary 49J20, 49M29
DOI:
https://doi.org/10.1090/qam/1955227
MathSciNet review:
MR1955227
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Abstract: A generalized Moreau-Yosida based primal-dual active set algorithm for the solution of a representative class of bilaterally control constrained optimal control problems with boundary control is developed. The use of the generalized Moreau-Yosida approximation allows an efficient identification of the active and inactive sets at each iteration level. The method requires no step-size strategy and exhibits a finite termination property for the discretized problem class. In infinite as well as in finite dimensions a convergence analysis based on an augmented Lagrangian merit function is given. In a series of numerical tests the efficiency of the new algorithm is emphasized.
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M. Bergounioux, K. Ito, and K. Kunisch, Primal-dual strategy for constrained optimal control problems, SIAM J. Control Optim. 37, 1176–1194 (1999)
M. Bergounioux and K. Kunisch, Augmented Lagrangian techniques for elliptic state constrained optimal control problems, SIAM J. Control Optim. 35, 1524–1543 (1997)
D. P. Bertsekas, Projected Newton methods for optimization problems with simple constraints, SIAM J. Control Optim. 20, 221–246 (1982)
M. Bergounioux, M. Haddou, M. Hintermüller, 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, 495–521 (2000)
R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology, Vol. 2, Springer-Verlag, 1988
J. E. Dennis, M. Heinkenschloss, and L. N. Vicente, TRICE: Trust-region interior-point SQP algorithms for optimal control and engineering design problems. http://www.caam.rice.edu/ trice
J. C. Dunn, On ${L^2}$ sufficient conditions and the gradient projection method for optimal control problems, SIAM J. Control Optim. 34, 1270–1290 (1996)
P. Gill, W. Murray, and M. H. Wright, Practical Optimization, Academic Press, 1999 (reprint)
D. Goldfarb and A. Idnani, A numerically stable dual method for solving strictly convex quadratic programs, Math. Prog. 21, 1–33 (1983)
P. Grisvard, Singularities in boundary value problems, Research Notes in Applied Mathematics 22, Springer-Verlag, 1992
W. W. Hager, The dual active set algorithm, Advances in Optimization and Parallel Computing, 1992, pp. 137–142
W. W. Hager, Application of the dual active set algorithm to quadratic network optimization, Comput. Optim. Appl. 1, 349–373 (1993)
M. Heinkenschloss, M. Ulbrich, and S. Ulbrich, Global convergence of trust-region interior-point algorithms for infinite-dimensional nonconvex minimization subject to pointwise bounds, preprint TR97-04, Dept. of Computational and Applied Mathematics, Rice University. Houston, 1997
K. Ito and K. Kunisch, Augmented Lagrangian-SQP methods in Hilbert spaces and applications to control in the coefficient problems, SIAM J. Optim. 6, 96–125 (1996)
K. Ito and K. Kunisch, Augmented Lagrangian methods for nonsmooth convex optimization in Hilbert spaces, Nonlinear Analysis, Theory, Methods and Applications 41, 591–616 (2000)
C. T. Kelley and E. W. Sachs, Multilevel algorithms for constrained compact fixed point problems. Iterative methods in numerical linear algebra, SIAM J. Sci. Comput. (3) 15, 645–667 (1994)
F. Leibfritz and E. W. Sachs, Inexact SQP interior point methods and large scale optimal control problems, preprint, University of Trier, 1995, SIAM J. Control Optim. 38, 272–293 (1999)
J. L. Lions, Some aspects of the optimal control of distributed parameter systems, Regional Conference Series in Applied Mathematics 6, SIAM, 1972
C. Lin and J. J. Moré, Newton’s method for large bound-constrained optimization problems, preprint, Argonne National Laboratory, Mathematics and Computer Science Division, 1999; SIAM J. Optim. 9, 1100–1127 (1999)
J. J. Moré and G. Toraldo, Algorithms for bound constrained quadratic programming problems, Numer. Math. 55, 377–400 (1989)
J. J. Moré and G. Toraldo, On the solution of large quadratic programming problems with bound constraints, SIAM J. Optim. 1 93–113 (1991)
T. Tian and J. C. Dunn, On the gradient projection method for optimal control problems with nonnegative ${L^2}$ inputs, SIAM J. Control Optim. 32, 516–537 (1994)
M. Tinkham, Introduction to Superconductivity, McGraw-Hill, N.Y., 1975
G. M. Troianiello, Elliptic differential equations and obstacle problems, Plenum Press, N.Y., 1987
F. Tröltzsch, An SQP-method for optimal control of a nonlinear heat equation, Control Cybernet. 23, 268–288 (1994)
R. J. Vanderbei, Linear programming: Foundations and extensions, Kluwer, 1997
S. J. Wright, Primal-dual interior point methods, SIAM, 1997
Y. Ye, Interior point algorithms: Theory and analysis, John Wiley and Sons, 1997
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