Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Differential dynamic programming and separable programs

  • 114 Accesses

  • 4 Citations


This paper deals with differential dynamic programming for solving nonlinear separable programs. The present algorithm and its derivation are rather different from differential dynamic programming algorithms and their derivations by Mayne and Jacobson, who have not proved the convergence of their algorithms. The local convergence of the present algorithm is proved, and numerical examples are given.

This is a preview of subscription content, log in to check access.


  1. 1.

    Mayne, D. Q., andPolak, E.,First-Order Strong Variation Algorithms for Optimal Control, Journal of Optimization Theory and Applications, Vol. 16, pp. 277–301, 1975.

  2. 2.

    Polak, E., andMayne, D. Q.,First-Order Strong Variation Algorithms for Optimal Control Problems with Terminal Inequality Constraints, Journal of Optimization Theory and Applications, Vol. 16, pp. 303–325, 1975.

  3. 3.

    Lasdon, L. S.,Optimization Theory for Large Systems, The Macmillan Company, New York, New York, 1970.

  4. 4.

    Hadley, G.,Nonlinear and Dynamic Programming, Addison-Wesley Publishing Company, Reading, Massachusetts, 1964.

  5. 5.

    Mayne, D. Q.,A Second-Order Gradient Method for Determining Optimal Trajectories of Non-Linear Discrete-Time Systems, International Journal of Control, Vol. 3, pp. 85–95, 1966.

  6. 6.

    Jacobson, D. H., andMayne, D. Q.,Differential Dynamic Programming, American Elsevier Publishing Company, New York, New York, 1970.

  7. 7.

    Gershwin, S. B., andJacobson, D. H.,A Discrete-Time Differential Dynamic Programming Algorithm with Application to Optimal Orbit Transfer, AIAA Journal, Vol. 8, pp. 1616–1626, 1970.

  8. 8.

    Havira, R. M., andLewis, J. B.,Computation of Quantized Controls Using Differential Dynamic Programming, IEEE Automatic Control, Vol. 17, pp. 191–196, 1972.

  9. 9.

    Mayne, D. Q.,Differential Dynamic Programming—A Unified Approach to the Optimization of Dynamic Systems, Control and Dynamic Systems, Vol. 10, Edited by C. T. Leondes, Academic Press, New York, New York, 1973.

  10. 10.

    Dyer, P., andMcReynolds, S. R.,The Computation and Theory of Optimal Control, Academic Press, New York, New York, 1970.

  11. 11.

    Fiacco, A. V., andMcCormick, G. P.,Nonlinear Programming: SUMT, John Wiley and Sons, New York, New York, 1968.

  12. 12.

    Ortega, J. M., andRheinboldt, W. C.,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.

  13. 13.

    Hearon, J. Z.,On the Singularity of a Certain Bordered Matrix, SIAM Journal on Applied Mathematics, Vol. 15, pp. 1413–1421, 1967.

  14. 14.

    Takahashi, I.,Variable Separation Principle for Mathematical Programming, Journal of the Operations Research Society of Japan, Vol. 6, pp. 82–105, 1964.

  15. 15.

    O'Neill, R. P., andWidhelm, W. B.,Computational Experience with Normed and Nonnormed Column-Generation Procedures in Nonlinear Programming, Operations Research, Vol. 23, pp. 372–382, 1975.

  16. 16.

    Tapia, R. A.,A Stable Approach to Newton's Method for General Mathematical Programming Problems in Rn, Journal of Optimization Theory and Applications, Vol. 14, pp. 453–476, 1974.

  17. 17.

    Miele, A., andLevy, A. V.,Modified Quasilinearization and Optimal Choice of the Multipliers, Part 1, Mathematical Programming Problems, Journal of Optimization Theory and Applications, Vol. 6, pp. 364–380, 1970.

  18. 18.

    Mine, H., andOhno, K.,Decomposition of Mathematical Programming Problems by Dynamic Programming and Its Application to Block-Diagonal Geometric Programs, Journal of Mathematical Analysis and Applications, Vol. 32, pp. 370–385, 1970.

  19. 19.

    Mine, H., Ohno, K., andFukushima, M.,Multilevel Decomposition of Nonlinear Programming Problems by Dynamic Programming, Journal of Mathematical Analysis and Applications, Vol. 53, pp. 7–27, 1976.

  20. 20.

    Kalman, R. E., andBertram, J. E.,Control System Analysis and Design Via the Second Method of Lyapunov, II, Discrete-Time System, Transactions of the ASME, Journal of Basic Engineering, Vol. 82, pp. 394–400, 1960.

Download references

Author information

Additional information

The author would like to express his appreciation to Professors H. Mine and T. Katayama for their helpful discussions. The author is also indebted to Professor D. Q. Mayne for drawing his attention to Refs. 1–2.

Communicated by D. Q. Mayne

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ohno, K. Differential dynamic programming and separable programs. J Optim Theory Appl 24, 617–637 (1978). https://doi.org/10.1007/BF00935303

Download citation

Key Words

  • Differential dynamic programming
  • nonlinear separable programs
  • convergence proofs
  • Kuhn-Tucker conditions
  • iterative methods
  • Newton's method