×

A nonsmooth Newton’s method for control-state constrained optimal control problems. (English) Zbl 1165.65361

Summary: The author investigates optimal control problems subject to mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer-Burmeister function the minimum principle is transformed into an equivalent nonlinear and nonsmooth equation in appropriate Banach spaces. This nonlinear and nonsmooth equation is solved by a nonsmooth Newton’s method. The local quadratic convergence is proved under certain regularity conditions and suggests a globalization strategy based on the minimization of the squared residual norm. A numerical example for the Rayleigh problem concludes the article.

MSC:

65K10 Numerical optimization and variational techniques
49J20 Existence theories for optimal control problems involving partial differential equations
49M25 Discrete approximations in optimal control
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Clarke, F. H., Optimization and Nonsmooth Analysis (1983), John Wiley & Sons: John Wiley & Sons New York · Zbl 0727.90045
[2] Fischer, A., A special Newton-type optimization method, Optimization, 24, 269-284 (1992) · Zbl 0814.65063
[3] Fischer, A., Solution of monotone complementarity problems with locally Lipschitzian functions, Math. Program., 76, 3, 513-532 (1997) · Zbl 0871.90097
[4] Gerdts, M., Direct Shooting Method for the Numerical Solution of Higher Index DAE Optimal Control Problems, Journal of Optimization Theory and Applications, 117, 2, 267-294 (2003) · Zbl 1033.65046
[5] M. Gerdts, Local minimum principle for optimal control problems subject to index one differential-algebraic equations, Technical Report, Department of Mathematics, University of Hamburg, http://www.math.unihamburg.de/home/gerdts/Report_index1.pdf; M. Gerdts, Local minimum principle for optimal control problems subject to index one differential-algebraic equations, Technical Report, Department of Mathematics, University of Hamburg, http://www.math.unihamburg.de/home/gerdts/Report_index1.pdf · Zbl 1123.49016
[6] Jiang, H., Global convergence analysis of the generalized Newton and Gauss-Newton methods of the Fischer-Burmeister equation for the complementarity problem, Math. Oper. Res., 24, 3, 529-543 (1999) · Zbl 0944.90094
[7] H. Maurer, D. Augustin, Sensitivity analysis and real-time control of parametric optimal control problems using boundary value methods, in: M. Grötschel, S.O. Krumke, J. Rambau, Herausgeber (Eds.), Online Optimization of Large Scale Systems, Springer, 2001, pp. 17-55.; H. Maurer, D. Augustin, Sensitivity analysis and real-time control of parametric optimal control problems using boundary value methods, in: M. Grötschel, S.O. Krumke, J. Rambau, Herausgeber (Eds.), Online Optimization of Large Scale Systems, Springer, 2001, pp. 17-55. · Zbl 0997.49024
[8] Neustadt, L. W., Optimization: A Theory of Necessary Conditions (1976), Princeton: Princeton New Jersey · Zbl 0166.09401
[9] Qi, L., Convergence analysis of some algorithms for solving nonsmooth equations, Mathematics of Operations Research, 18, 1, 227-244 (1993) · Zbl 0776.65037
[10] Qi, L.; und Sun, J., A nonsmooth version of Newton’s method, Mathematical Programming, 58, 3, 353-367 (1993) · Zbl 0780.90090
[11] Ulbrich, M., Nonsmooth Newton-like Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. Habilitation (2002), Technical University of Munich: Technical University of Munich Munich
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.