Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



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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  • [1] M. Bergounioux, K. Ito, and K. Kunisch, Primal-dual strategy for constrained optimal control problems, SIAM J. Control Optim. 37, 1176-1194 (1999) MR 1691937
  • [2] M. Bergounioux and K. Kunisch, Augmented Lagrangian techniques for elliptic state constrained optimal control problems, SIAM J. Control Optim. 35, 1524-1543 (1997) MR 1466914
  • [3] D. P. Bertsekas, Projected Newton methods for optimization problems with simple constraints, SIAM J. Control Optim. 20, 221-246 (1982) MR 646950
  • [4] 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) MR 1787272
  • [5] R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology, Vol. 2, Springer-Verlag, 1988 MR 969367
  • [6] 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
  • [7] 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) MR 1395833
  • [8] P. Gill, W. Murray, and M. H. Wright, Practical Optimization, Academic Press, 1999 (reprint) MR 634376
  • [9] D. Goldfarb and A. Idnani, A numerically stable dual method for solving strictly convex quadratic programs, Math. Prog. 21, 1-33 (1983)
  • [10] P. Grisvard, Singularities in boundary value problems, Research Notes in Applied Mathematics 22, Springer-Verlag, 1992 MR 1173209
  • [11] W. W. Hager, The dual active set algorithm, Advances in Optimization and Parallel Computing, 1992, pp. 137-142
  • [12] W. W. Hager, Application of the dual active set algorithm to quadratic network optimization, Comput. Optim. Appl. 1, 349-373 (1993) MR 1216720
  • [13] 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
  • [14] 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) MR 1377727
  • [15] 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) MR 1780634
  • [16] 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) MR 1273157
  • [17] 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) MR 1740596
  • [18] J. L. Lions, Some aspects of the optimal control of distributed parameter systems, Regional Conference Series in Applied Mathematics 6, SIAM, 1972 MR 0479526
  • [19] 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) MR 1724778
  • [20] J. J. Moré and G. Toraldo, Algorithms for bound constrained quadratic programming problems, Numer. Math. 55, 377-400 (1989) MR 997229
  • [21] J. J. Moré and G. Toraldo, On the solution of large quadratic programming problems with bound constraints, SIAM J. Optim. 1 93-113 (1991) MR 1094793
  • [22] 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) MR 1261152
  • [23] M. Tinkham, Introduction to Superconductivity, McGraw-Hill, N.Y., 1975
  • [24] G. M. Troianiello, Elliptic differential equations and obstacle problems, Plenum Press, N.Y., 1987 MR 1094820
  • [25] F. Tröltzsch, An SQP-method for optimal control of a nonlinear heat equation, Control Cybernet. 23, 268-288 (1994) MR 1284519
  • [26] R. J. Vanderbei, Linear programming: Foundations and extensions, Kluwer, 1997
  • [27] S. J. Wright, Primal-dual interior point methods, SIAM, 1997 MR 1422257
  • [28] Y. Ye, Interior point algorithms: Theory and analysis, John Wiley and Sons, 1997 MR 1481160

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DOI: https://doi.org/10.1090/qam/1955227
Article copyright: © Copyright 2003 American Mathematical Society

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