A Structured Interior Point SQP Method for Nonlinear Optimal Control Problems

  • Marc C. Steinbach
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 115)


Direct boundary value problem methods in connection with SQP iteration have proven very successful in solving nonlinear optimal control problems. Such methods use parameterized control functions, discretize the state differential equations by, e.g., multiple shooting or collocation, and treat the discretized BVP as an equality-constraint in a large nonlinear constrained optimization problem. In realistic applications several thousands of variables can appear in the NLP. Solution by standard techniques is therefore impractical. A careful choice of the discretization leads to QP subproblems possessing a very special m-stage block-sparse structure, where m is the grid size. The paper presents a recursive solution algorithm that fully exploits this QP sparseness to generate a factorization of the inverse of the KKT matrix in O(m) operations. A structure-preserving primal-dual barrier method is proposed for treating the generally large number of state and control inequality constraints.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. T. Betts, W. P. Huffman: Path Constrained Trajectory Optimization Using Sparse Sequential Quadratic Programming, Boeing Computer Services, 1991.Google Scholar
  2. [2]
    H. G. Bock, K.-J. Plitt: A Multiple Shooting Algorithm for Direct Solution of Optimal Control Processes, Proc. 9th IFAC World Congress, Budapest, 1984.Google Scholar
  3. [3]
    A. V. Fiacco, G. P McCormick.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley and Sons, New York, 1968.zbMATHGoogle Scholar
  4. [4]
    K. R. Frisch: The Logarithmic Potential Method of Convex Programming, Unpublished manuscript, University Institue of Economics, Oslo, 1955.Google Scholar
  5. [5]
    P. E. Gill, W. Murray, M. A. Saunders, M. H. Wright: A Schur-Complement Method for Sparse Quadratic Programming, Report SOL 87–12, Department of Operations Research, Stanford University.Google Scholar
  6. [6]
    P. E. Gill, W. Murray, D. B. Ponceleón, M. H. Wright: Solving Reduced KKT Systems in Barrier Methods for Linear and Quadratic Programming, Report SOL 91–7, Department of Operations Research, Stanford University.Google Scholar
  7. [7]
    C. Gonzaga: An Interior Trust Region Method for Linearly Constrained Optimization, MPS Newsletter 19 (1991) 55–65.Google Scholar
  8. [8]
    C. R. Hargraves, R. W. Paris: Direct Trajectory Optimization Using Nonlinear Programming and Collocation, AIAA J. Guidance 10 (1987) 338–342.zbMATHCrossRefGoogle Scholar
  9. [9]
    N. Karmarkar: A New Polynomial Time Algorithm for Linear Programming, Combinatorica 4 (1984) 373–395.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    J. Konzelmann, H. G. Bock, R. W. Longman: Time Optimal Trajectories of Polar Robot Manipulators by Direct Methods, Modeling and Simulation, 20/5 (1989) 1933–1939, Instrument Society of America.Google Scholar
  11. [11]
    J. Konzelmann, H. G. Bock, R. W. Longman: Time Optimal Trajectories of Elbow Robots by Direct Methods, Proc. AIAA Guidance, Navigation and Control Conference, Boston (1989) AIAA Paper 89–3530-CP.Google Scholar
  12. [12]
    S. Mehrotra: On the Implementation of a (Primal-Dual) Interior Point Method, Technical Report 90–03, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, 1990.Google Scholar
  13. [13]
    R. D. C. Monteiro, I. Adler: Interior Path Following Primal-Dual Algorithms. Part II: Convex Quadratic Programming, Mathematical Programming 44 (1989) 43–66.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    K.-J. Plitt: Ein superlinear konvergentes Mehrzielverfahren zur direkten Berechnung beschränkter optimaler Steuerungen, Diploma Thesis (in German), Department of Applied Mathematics, University of Bonn, 1981.Google Scholar
  15. [15]
    M. C. Steinbach, H. G. Bock, R. W. Longman: Time Optimal Control of SCARA Robots, Proc. AIAA Guidance, Navigation and Control Conference, Portland (1990) AIAA Paper 90-3394-CP.Google Scholar
  16. [16]
    O. von Stryk: Numerical Solution of Optimal Control Problems by Direct Collocation, Report No. 322, Department of Mathematics, Munich University of Technology, 1991.Google Scholar
  17. [17]
    Y. Ye, E. Tse: An Extension of Karmarkar’s Projective Algorithm for Convex Quadratic Programming, Mathematical Programming 44 (1989) 157–179.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel 1994

Authors and Affiliations

  • Marc C. Steinbach
    • 1
  1. 1.Interdisciplinary Center for Scientific ComputationUniversity of HeidelbergHeidelbergGermany

Personalised recommendations