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Solving optimal control problems with terminal complementarity constraints via Scholtes’ relaxation scheme

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We investigate the numerical treatment of optimal control problems of linear ordinary differential equations with terminal complementarity constraints. Therefore, we generalize the well-known relaxation technique of Scholtes to the problem at hand. In principle, any other relaxation approach from finite-dimensional complementarity programming can be adapted in similar fashion. It is shown that the suggested method possesses strong convergence properties under mild assumptions. Finally, some numerical examples are presented.

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References

  1. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Elsevier, Kidlington (2003)

    MATH  Google Scholar 

  2. Agarwal, R.P., O’Regan, D.: An Introduction to Ordinary Differential Equations. Springer, New York (2008)

    Book  MATH  Google Scholar 

  3. Barnett, S., Cameron, R.G.: Introduction to Mathematical Control Theory. Oxford University Press, New York (1990)

    MATH  Google Scholar 

  4. Beard, R.W., Lawton, J., Hadaegh, F.Y.: A coordination architecture for spacecraft formation control. IEEE Trans. Control Syst. Technol. 9(6), 777–790 (2001). https://doi.org/10.1109/87.960341

    Article  Google Scholar 

  5. Benita, F., Mehlitz, P.: Optimal control problems with terminal complementarity constraints. SIAM J. Optim. 28(4), 3079–3104 (2018). https://doi.org/10.1137/16M107637X

    Article  MathSciNet  MATH  Google Scholar 

  6. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2002)

    MATH  Google Scholar 

  7. Bonuccelli, M.A., Martelli, F., Pelagatti, S.: Optimal packet scheduling in tree-structured LEO satellite clusters. Mobile Netw. Appl. 9(4), 289–295 (2004). https://doi.org/10.1145/1023663.1023674

    Article  Google Scholar 

  8. Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, Chichester (2016)

    Book  MATH  Google Scholar 

  9. Dontchev, A.L.: Discrete approximations in optimal control. In: Mordukhovich, B.S., Sussmann, H.J. (eds.) Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control, pp. 59–80. Springer, New York (1996)

    Chapter  Google Scholar 

  10. Gerdts, M.: Optimal Control of ODEs and DAEs. De Gruyter, Berlin (2012)

    Book  MATH  Google Scholar 

  11. Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30(1), 45–61 (2005). https://doi.org/10.1007/s10589-005-4559-5

    Article  MathSciNet  MATH  Google Scholar 

  12. Hinze, M., Rösch, A.: Discretization of Optimal Control Problems, pp. 391–430. Springer, Basel (2012). https://doi.org/10.1007/978-3-0348-0133-1_21

    Book  MATH  Google Scholar 

  13. Hoheisel, T., Kanzow, C., Schwartz, A.: Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints. Math. Program. Ser. A 137(1), 257–288 (2013). https://doi.org/10.1007/s10107-011-0488-5

    Article  MathSciNet  MATH  Google Scholar 

  14. Jahn, J.: Introduction to the Theory of Nonlinear Optimization. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  15. Kalashnikov, V.V., Benita, F., Mehlitz, P.: The natural gas cash-out problem: a bilevel optimal control approach. Math. Probl. Eng. (2015). https://doi.org/10.1155/2015/286083

  16. Löber, J.: Optimal Trajectory Tracking of Nonlinear Dynamical Systems. Springer, Berlin (2017)

    Book  MATH  Google Scholar 

  17. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  18. Olfati-Saber, R.: Flocking for multi-agent dynamic systems: algorithms and theory. IEEE Trans. Autom. Control. 51(3), 401–420 (2006). https://doi.org/10.1109/TAC.2005.864190

    Article  MathSciNet  MATH  Google Scholar 

  19. Outrata, J.V., Kočvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer, Dordrecht (1998)

    Book  MATH  Google Scholar 

  20. Robinson, S.M.: Stability theory for systems of inequalities, part II: differentiable nonlinear systems. SIAM J. Numer. Anal. 13(4), 497–513 (1976). https://doi.org/10.1137/0713043

    Article  MathSciNet  MATH  Google Scholar 

  21. Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11(4), 918–936 (2001). https://doi.org/10.1137/S1052623499361233

    Article  MathSciNet  MATH  Google Scholar 

  22. Schwartz, A., Polak, E.: Consistent approximations for optimal control problems based on Runge–Kutta integration. SIAM J. Control Optim. 34(4), 1235–1269 (1996). https://doi.org/10.1137/S0363012994267352

    Article  MathSciNet  MATH  Google Scholar 

  23. Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. Ser. A 106(1), 25–57 (2006). https://doi.org/10.1007/s10107-004-0559-y

    Article  MathSciNet  MATH  Google Scholar 

  24. Zowe, J., Kurcyusz, S.: Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5(1), 49–62 (1979). https://doi.org/10.1007/BF01442543

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous reviewers for some valuable comments which helped us to improve the presentation of our results.

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  1. Singapore University of Technology and Design, Singapore, 487372, Singapore

    Francisco Benita

  2. Chair of Optimal Control, Brandenburgische Technische Universität Cottbus-Senftenberg, 03046, Cottbus, Germany

    Patrick Mehlitz

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  1. Francisco Benita
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  2. Patrick Mehlitz
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Correspondence to Patrick Mehlitz.

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Benita, F., Mehlitz, P. Solving optimal control problems with terminal complementarity constraints via Scholtes’ relaxation scheme. Comput Optim Appl 72, 413–430 (2019). https://doi.org/10.1007/s10589-018-0050-y

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