A quadratically convergent inexact SQP method for optimal control of differential algebraic equations

SUMMARY In this paper, we present an inexact sequential quadratic programming method in the context of a direct multiple shooting approach for differential algebraic equations. For the case that a numerical integration routine is used to compute the states of a relaxed differential algebraic equation, the computation of sensitivities, with respect to a large number of algebraic states, can become very expensive. To overcome this limitation, the inexact sequential quadratic programming method that we propose in this paper requires neither the computation of any sensitivity direction of the differential state trajectory, with respect to the algebraic states, nor the consistent initialization of the differential algebraic equation. We prove the locally quadratic convergence of the proposed method. Finally, we demonstrate the numerical performance of the method by optimizing a distillation column with 82 differential and 122 algebraic states. Copyright © 2012 John Wiley & Sons, Ltd.

[1]  Moritz Diehl,et al.  The Lifted Newton Method and Its Application in Optimization , 2009, SIAM J. Optim..

[2]  H. Maurer,et al.  SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control , 2000 .

[3]  S. Miesbach,et al.  Optimal Temperature Control of Semibatch Polymerization Reactors , 1996 .

[4]  Andrea Walther,et al.  Numerical experiments with an inexact Jacobian trust-region algorithm , 2011, Comput. Optim. Appl..

[5]  Andrea Walther,et al.  An adjoint-based SQP algorithm with quasi-Newton Jacobian updates for inequality constrained optimization , 2010, Optim. Methods Softw..

[6]  Dieter Kraft,et al.  On Converting Optimal Control Problems into Nonlinear Programming Problems , 1985 .

[7]  Moritz Diehl,et al.  Real-Time Optimization for Large Scale Nonlinear Processes , 2001 .

[8]  Paul T. Boggs,et al.  Sequential Quadratic Programming , 1995, Acta Numerica.

[9]  G. R. Sullivan,et al.  The development of an efficient optimal control package , 1978 .

[10]  H. Bock,et al.  A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems , 1984 .

[11]  Moritz Diehl,et al.  ACADO toolkit—An open‐source framework for automatic control and dynamic optimization , 2011 .

[12]  Irene Bauer.  Numerische Verfahren zur Lösung von Anfangswertaufgaben und zur Generierung von ersten und zweiten Ableitungen mit Anwendungen bei Optimierungsaufgaben in Chemie und Verfahrenstechnik , 1999 .

[13]  W. Rheinboldt.  Differential-algebraic systems as differential equations on manifolds , 1984 .

[14]  C F Curtiss,et al.  Integration of Stiff Equations. , 1952, Proceedings of the National Academy of Sciences of the United States of America.