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Journal of Optimization Theory and Applications

, Volume 180, Issue 2, pp 518–535 | Cite as

On Lifting Operators and Regularity of Nonsmooth Newton Methods for Optimal Control Problems of Differential Algebraic Equations

  • Jinhai ChenEmail author
  • Herschel Rabitz
Article
  • 46 Downloads

Abstract

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.

Keywords

Optimal control of DAEs Nonsmooth Newton methods Lifting operators Regularity conditions Convergence 

Mathematics Subject Classification

49J15 49J52 49M15 

Notes

Acknowledgements

JC is financially supported by DOE (Grant No. DE-FG02-02ER15344). HR is financially supported by NSF (Grant No. CHE-1763198) and DOE (Grant No. DE-FG02-02ER15344).

References

  1. 1.
    Biegler, L.T.: Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes. MOS/SIAM Series on Optimization, vol. 10. SIAM, Philadelphia (2010)CrossRefzbMATHGoogle Scholar
  2. 2.
    Dontchev, A.L., Hager, W.W., Malanowski, K.: Error bounds for Euler approximation of a state and control constrained optimal control problem. Numer. Funct. Anal. Optim. 21, 653–682 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dontchev, A.L., Hager, W.W., Veliov, V.M.: Second-order Runge-Kutta approximations in control constrained optimal control. SIAM J. Numer. Anal. 38, 202–226 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hager, W.W.: Runge-Kutta methods in optimal control and the transformed adjoint system. Numer. Math. 87, 247–282 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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)Google Scholar
  6. 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)Google Scholar
  7. 7.
    Gerdts, M.: Optimal control of ODEs and DAEs de Gruyter Textbook. Walter de Gruyter & Co., Berlin (2012)CrossRefzbMATHGoogle Scholar
  8. 8.
    Büskens, C.: Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustandsbeschränkungen, PhD thesis, Westfälische Wilhems-Universität Münster (1998)Google Scholar
  9. 9.
    Grötschel, M., Krumke, S.O., Rambau, J.: Online Optimization of Large Scale Systems. Springer, Berlin (2001)CrossRefzbMATHGoogle Scholar
  10. 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)zbMATHGoogle Scholar
  11. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hartl, R.F., Sethi, S.P., Vickson, G.: A survey of the maximum principles for optimal control problems with state constraints. SIAM Rev. 37, 181–218 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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)Google Scholar
  14. 14.
    Pesch, H.J.: A practical guide to the solution of real-life optimal control problems. Control Cybern. 23, 7–60 (1995)MathSciNetzbMATHGoogle Scholar
  15. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kanzow, C., Pieper, H.: Jacobian smoothing methods for nonlinear complementarity problems. SIAM J. Optim. 9, 342–372 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Chen, B., Harker, P.T.: A non-interior-point continuation method for linear complementarity problem. SIAM J. Matrix Anal. Appl. 14, 1168–1190 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Chen, X., Nashed, Z., Qi, L.: Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38, 1200–1216 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)zbMATHGoogle Scholar
  24. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ulbrich, M.: Semismooth newton methods for operator equations in function spaces. SIAM J. Optim. 13, 805–841 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ulbrich, M.: Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces, vol. 11. SIAM, Philadelphia (2011)CrossRefzbMATHGoogle Scholar
  27. 27.
    Tröltzsch, F.: Regular Lagrange multipliers for control problems with mixed pointwise control state constraints. SIAM J. Optim. 15, 616–634 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Alt, W., Malanowski, K.: The Lagrange–Newton method for nonlinear optimal control problems. Comput. Optim. Appl. 2, 77–100 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Malanowski, K.: On normality of Lagrange multipliers for state constrained optimal control problems. Optimization 52, 75–91 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  32. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control, 15. SIAM, Philadelphia (2008)zbMATHGoogle Scholar
  34. 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)Google Scholar
  35. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Malanowski, K., Maurer, H.: Sensitivity analysis for parametric control problems with control-state constraints. Comput. Optim. Appl. 5, 253–283 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 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)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Boston UniversityBostonUSA
  2. 2.Princeton UniversityPrincetonUSA

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