Global convergence of a nonsmooth Newton method for control-state constrained optimal control problems

SIAM Journal on Optimization

Volumn 19, Issue 1, 2008, Pages 326-350

  Gerdts, Matthias ^a  

^a UNIVERSITY OF BIRMINGHAM   (United Kingdom)

Author keywords

Control state constraints; Global convergence; Nonsmooth Newton method; Optimal control

Indexed keywords

BANACH SPACES; NONLINEAR EQUATIONS; OPTIMIZATION; TOPOLOGY;

CONSTRAINED OPTIMAL CONTROL PROBLEMS; CONTROL-STATE CONSTRAINTS; FISCHER-BURMEISTER FUNCTIONS; GLOBAL CONVERGENCE; LOCAL MINIMUMS; MIXED CONTROLS; NON-SMOOTH EQUATIONS; NONSMOOTH NEWTON METHOD; NONSMOOTH NEWTON'S METHODS; NUMERICAL EXAMPLES; NUMERICAL SOLUTIONS; OPTIMAL CONTROL; OPTIMAL CONTROL PROBLEMS; RAYLEIGH; REGULARITY CONDITIONS; RESIDUAL NORMS; SUPERLINEAR CONVERGENCES;

NEWTON-RAPHSON METHOD;

EID: 61349167886     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/060657546     Document Type: Article

References (34)

1. W. ALT AND K. MALANOWSKI, The Lagrange-Newton method for nonlinear optimal control problems, Comput. Optim. Appl., 2 (1993), pp. 77-100.
2. W. ALT AND K. MALANOWSKI, The Lagrange-Newton method for state constrained optimal control problems, Comput. Optim. Appl., 4 (1995), pp. 217-239.
3. C. BÜSKENS, Optimierungsmethoden und Sensitivitätsanalyse für Optimale Steuerprozesse mit Steuer- und Zustandsbeschränkungen, Ph.D. thesis, Fachbereich Mathematik, Westfälische Wilhems-Universität Münster, Münster, Germany, 1998.
4. X. CHEN, Z. NASHED, AND L. QI, Smoothing methods and semismooth methods for nondifferentiable operator equations, SIAM J. Numer. Anal., 38 (2000), pp. 1200-1216.
5. F. H. CLARKE, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983.
6. A. L. DONTCHEV, W. W. HAGER, AND K. MALANOWSKI, Error bounds for Euler approximation of a state and control constrained optimal control problem, Numer. Funct. Anal. Optim., 21 (2000), pp. 653-682.
7. A. L. DONTCHEV, W. W. HAGER, AND V. M. VELIOV, Second-order Runge-Kutta approximations in control constrained optimal control, SIAM J. Numer. Anal., 38 (2000), pp. 202-226.
8. A. FISCHER, A special Newton-type optimization method, Optimization, 24 (1992), pp. 269-284.
9. A. FISCHER, Solution of monotone complementarity problems with locally Lipschitzian functions, Math. Programming., 76 (1997), pp. 513-532.
10. M. GERDTS, Direct shooting method for the numerical soluti-of higher index DAE optimal control problems, J. Optim. Theory Appl., 117 (2003), pp. 267-294.
11. M. GERDTS, A nonsmooth Newton's method for control-state constrained optimal control problems, in Proceedings of the 5th Annual Vienna Symposium on Mathematical Modelling, ARGESIM Reports 30, Vienna, 2006.
12. M. GERDTS, Optimal Control of Ordinary Differential Equations and Differential-Algebraic Equations, Habilitation thesis, Universität Bayreuth, Bayreuth, 2006; also available online from http://web.mat.bham.ac.uk/ M.Gerdts/habilitation.pdf.
13. M. GRÖTSCHEL, S. O. KRUMKE, AND J. RAMBAU. EDS., Online Optimization of Large Scale Systems, Springer, Berlin, 2001.
14. W. W. HAGER, The dual active set algorithm and its application to linear programming, SIAM J. Control Optim., 17 (1979), pp. 321-338.
15. W. W. HAGER, Runge-Kutta methods in optimal control and the transformed adjoint system, Numer. Math., 87 (2000), pp. 247-282.
16. R. F. HARTL, S. P. SETHI, AND G. VICKSON, A survey of the maximum principles for optimal control problems with state constraints, SIAM Rev., 37 (1995), pp. 181-218.
17. A. D. IOFFE AND V. M. TIHOMIROV, Theory of Extremal Problems, in Stud. Math. Appl. 6, North-Holland, Amsterdam, 1979.
18. H. JIANG, Global convergence analysis of the generalized Newton and Gauss-Newton methods of the Fischer-Burmeister equation for the complementarity problem, Math. Oper. Res., 24 (1999), pp. 529-543.
19. B. KUMMER, Newton's method for non-differentiable functions, in Advances in Mathematical Optimization, J. Guddat et al., eds., Akademie-Verlag, Berlin, 1988, pp. 171-194.
20. B. KUMMER, Newton's method based on generalized derivatives for nonsmooth functions: Convergence analysis, in Advances in Optimization, W. Oettli and D. Pallaschke, eds., 1991, Springer, Berlin, pp. 171-194.
21. L. A. LJUSTERNIK AND W. I. SOBOLEW, Elemente der Funktionalanalysis, Verlag Harri Deutsch, Zürich, 1976.
22. K. MALANOWSKI, On normality of Lagrange multipliers for state constrained optimal control problems, Optimization, 52 (2003), pp. 75-91.
23. K. MALANOWSKI, C. BÜSKENS, AND H. MAURER, Convergence of approximations to nonlinear optimal control problems, in Mathematical Programming with Data Perturbations, Anthony Fiacco, ed., Lecture Notes in Pure and Appl. Math. 195, Marcel Dekker, New York, 1997, pp. 253-284.
24. K. MALANOWSKI AND H. MAURER, Sensitivity analysis for parametric control problems with control-state constraints, Comput. Optim. Appl., 5 (1996), pp. 253-283.
25. K. MALANOWSKI, H. MAURER, AND S. PICKENHAIN, Second-order sufficient conditions for state-constrained optimal control problems, J. Optim. Theory Appl., 123 (2004), pp. 595-617.
26. H. MAURER AND D. AUGUSTIN, Sensitivity Analysis and Real-Time Control of Parametric Optimal Control Problems Using Boundary Value Methods, in Online Optimization of Large Scale Systems, M. Grötschel, S. O. Krumke, and J. Rambau, eds., Springer, New York, 2001, pp. 17-55.
27. L. W. NEUSTADT, Optimization: A Theory of Necessary Conditions, Princeton University Press, Princeton, NJ, 1976.
28. H. J. OBERLE AND W. GRIMM, Bndsco - A Program for the Numerical Solution of Optimal Control Problems, Tech. Report Reihe B, Bericht 36, Hamburger Beiträge zur Angewandten Mathematik, Department of Mathematics, University of Hamburg, Hamburg, Germany; also available online from http://www.math.uni-hamburg.de/home/oberle/software.html, 2001.
29. L. QI, Convergence analysis of some algorithms for solving nonsmooth equations, Math. Oper. Res., 18 (1993), pp. 227-244.
30. L. QI AND J. SUN, A nonsmooth version of Newton's method, Math. Program., 58 (1993), pp. 353-367.
31. M. ULBRICH, Nonsmooth Newton-Like Methods fVariational Inequalities and Constrained Optimization Problems in Function Spaces, Habilitation, Technical University of Munich, Munich, Germany, 2002.
32. M. ULBRICH, Semismooth Newton methods for operator equations in function spaces, SIAM J. Optim., 13 (2003), pp. 805-841.
33. D. WERNER, Funktionalanalysis, Springer, Berlin, 1995.
34. V. ZEIDAN, The Riccati equation for optimal control problems with mixed state-control constraints: Necessity and sufficiency, SIAM J. Control Optim., 32 (1994), pp. 1297-1321.