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On lifting operators and regularity of nonsmooth Newton methods for optimal control problems of differential algebraic equations. (English) Zbl 1409.49004
Summary: This paper focuses on nonsmooth Newton methods of optimal control problems governed by mixed control-state constraints with differential algebraic equations. In contrast to previous results, we analyze lifting operators involved in nonsmooth Newton methods and establish corresponding convergence results. We also give sufficient conditions for regularity of generalized derivatives of systems of nonsmooth operator equations associated with optimal control problems.
Reviewer: Reviewer (Berlin)
49J15 Existence theories for optimal control problems involving ordinary differential equations
49J52 Nonsmooth analysis
49M15 Newton-type methods
Full Text: DOI
[1] Biegler, L.T.: Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes. MOS/SIAM Series on Optimization, vol. 10. SIAM, Philadelphia (2010) · Zbl 1207.90004
[2] Dontchev, AL; Hager, WW; Malanowski, K., Error bounds for Euler approximation of a state and control constrained optimal control problem, Numer. Funct. Anal. Optim., 21, 653-682, (2000) · Zbl 0969.49013
[3] Dontchev, AL; Hager, WW; Veliov, VM, Second-order Runge-Kutta approximations in control constrained optimal control, SIAM J. Numer. Anal., 38, 202-226, (2000) · Zbl 0968.49022
[4] Hager, WW, Runge-Kutta methods in optimal control and the transformed adjoint system, Numer. Math., 87, 247-282, (2000) · Zbl 0991.49020
[5] Leineweber, D.B., Bock, H.G., Schlöder, J.P., Gallitzendörfer, J.V., Schäfer, A., Jansohn, P.: A Boundary Value Problem Approach to the Optimization of Chemical Processes Described by DAE Models. University of Heidelberg, Technical Report, Interdisciplinary Center for Scientific Computing (1997)
[6] Malanowski, K., Büskens, C., Maurer, H.: Convergence of Approximations to Nonlinear Optimal Control Problems, Mathematical Programming with Data Perturbations, Lecture Notes in Pure and Appl. Math., Dekker, New York, vol. 195, pp. 253-284 (1998)
[7] Gerdts, M.: Optimal control of ODEs and DAEs de Gruyter Textbook. Walter de Gruyter & Co., Berlin (2012) · Zbl 1275.49001
[8] Büskens, C.: Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustandsbeschränkungen, PhD thesis, Westfälische Wilhems-Universität Münster (1998)
[9] Grötschel, M., Krumke, S.O., Rambau, J.: Online Optimization of Large Scale Systems. Springer, Berlin (2001) · Zbl 0971.00007
[10] Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems, Studies in Mathematics and its Applications, 6. North-Holland Publishing Co., Amsterdam, New York (1979) · Zbl 0407.90051
[11] Gerdts, M., Direct shooting method for the numerical solution of higher index DAE optimal control problems, J. Optim. Theory Appl., 117, 267-294, (2003) · Zbl 1033.65046
[12] Hartl, RF; Sethi, SP; Vickson, G., A survey of the maximum principles for optimal control problems with state constraints, SIAM Rev., 37, 181-218, (1995) · Zbl 0832.49013
[13] Oberle, H.J., Grimm, W.: Bndsco—A Program for the Numerical Solution of Optimal Control Problems. Institute for Flight Systems Dynamics, DLR, Oberpfaffenhofen, Internal Report 515-89/22 (1989)
[14] Pesch, HJ, A practical guide to the solution of real-life optimal control problems, Control Cybern., 23, 7-60, (1995) · Zbl 0811.49029
[15] Chen, J.; Gerdts, M., Numerical solution of control-state constrained optimal control problems with an inexact smoothing Newton method, IMA J. Numer. Anal., 31, 1598-1624, (2011) · Zbl 1242.65122
[16] Chen, J.; Gerdts, M., Smoothing techniques of nonsmooth Newton methods for control-state constrained optimal control problems, SIAM J. Numer. Anal., 50, 1982-2011, (2012) · Zbl 1252.49045
[17] Kanzow, C.; Pieper, H., Jacobian smoothing methods for nonlinear complementarity problems, SIAM J. Optim., 9, 342-372, (1999) · Zbl 0986.90065
[18] Qi, L.; Sun, D., Smoothing functions and a smoothing Newton method for complementarity and variational inequality problems, J. Optim. Theory Appl., 113, 121-147, (2002) · Zbl 1032.49017
[19] Chen, B.; Harker, PT, A non-interior-point continuation method for linear complementarity problem, SIAM J. Matrix Anal. Appl., 14, 1168-1190, (1993) · Zbl 0788.65073
[20] Chen, J.; Qi, L., Globally and superlinearly convergent inexact Newton-Krylov algorithms for solving nonsmooth equations, Numer. Linear Algebra Appl., 17, 155-174, (2010) · Zbl 1240.65168
[21] Chen, X.; Nashed, Z.; Qi, L., Smoothing methods and semismooth methods for nondifferentiable operator equations, SIAM J. Numer. Anal., 38, 1200-1216, (2000) · Zbl 0979.65046
[22] Chen, X.; Qi, L.; Sun, D., Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities, Math. Comput., 67, 519-540, (1998) · Zbl 0894.90143
[23] Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003) · Zbl 1062.90001
[24] Qi, L.; Sun, D.; Zhou, G., A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Math. Program., 87, 1-35, (2000) · Zbl 0989.90124
[25] Ulbrich, M., Semismooth newton methods for operator equations in function spaces, SIAM J. Optim., 13, 805-841, (2003) · Zbl 1033.49039
[26] Ulbrich, M.: Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces, vol. 11. SIAM, Philadelphia (2011) · Zbl 1235.49001
[27] Tröltzsch, F., Regular Lagrange multipliers for control problems with mixed pointwise control state constraints, SIAM J. Optim., 15, 616-634, (2005) · Zbl 1083.49018
[28] Alt, W.; Malanowski, K., The Lagrange-Newton method for nonlinear optimal control problems, Comput. Optim. Appl., 2, 77-100, (1993) · Zbl 0774.49022
[29] Malanowski, K., On normality of Lagrange multipliers for state constrained optimal control problems, Optimization, 52, 75-91, (2003) · Zbl 1057.49004
[30] Zeidan, V., The Riccati equation for optimal control problems with mixed state-control constraints: necessity and sufficiency, SIAM J. Control Optim., 32, 1297-1321, (1994) · Zbl 0810.49024
[31] Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) · Zbl 0582.49001
[32] Hintermüller, M.; Ito, K.; Kunisch, K., The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim., 13, 865-888, (2003) · Zbl 1080.90074
[33] Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control, 15. SIAM, Philadelphia (2008) · Zbl 1156.49002
[34] Gerdts, M.: Global convergence of a nonsmooth Newton method for control-state constrained optimal control problems, SIAM J. Optim. 19, 326-350 (2008), Erratum 21, 615-616 (2011) · Zbl 1158.49031
[35] Gerdts, M.; Kunkel, M., A globally convergent semi-smooth Newton method for control-state constrained DAE optimal control problems, Comput. Optim. Appl., 48, 601-633, (2011) · Zbl 1226.49021
[36] Malanowski, K.; Maurer, H., Sensitivity analysis for parametric control problems with control-state constraints, Comput. Optim. Appl., 5, 253-283, (1996) · Zbl 0864.49020
[37] Maurer, H.; Pickenhain, S., Second-order sufficient conditions for control problems with mixed control-state constraints, J. Optim. Theory Appl., 86, 649-667, (1995) · Zbl 0874.49020
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