Journal of Optimization Theory and Applications

, Volume 47, Issue 2, pp 195–216 | Cite as

The role of the multipliers in the multiplier method

  • M. A. Ibiejugba
Contributed Papers

Abstract

The paper studies the role of the multipliers when the multiplier method is applied as a computational technique for minimizing penalized cost functionals for optimal control problems characterized by linear systems and integral quadratic costs.

Key Words

Multiplier methods unconstrained problems Lagrange multipliers penalty function methods convergence rate 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Glad, T.,Constrained Optimization Using Multiplier Methods with Applications to Control Problems, University of Lund, Lund, Sweden, Report No. 7603, 1976.Google Scholar
  2. 2.
    Ibiejugba, M. A.,Computing Methods in Optimal Control, University of Leeds, Leeds, England, PhD Thesis, 1980.Google Scholar
  3. 3.
    Ibiejugba, M. A.,On Ritz-Penalty Method for Solving the Control of a Diffusion Equation, Journal of Optimization Theory and Applications, Vol. 39, pp. 431–449, 1983.Google Scholar
  4. 4.
    Ibiejugba, M. A., andOlufeagba, B. J.,Conventional Penalty Optimization Methods, Journal of Optimization Theory and Applications (in press).Google Scholar
  5. 5.
    Aderibigbe, F. M., Ibiejugba, M. A., andOnumanyi, P.,On Application of a Control Operator to Delay Equations, Journal of Mathematical Analysis and Applications (to appear).Google Scholar
  6. 6.
    Di Pillo, G., andGrippo, L.,A Computing Algorithm for the Epsilon Method to Identification and Optimal Control Problems, Ricerche di Automatica, Vol. 3, pp. 54–77, 1972.Google Scholar
  7. 7.
    Pierre, D. A., andLowe, M. J.,Mathematical Programming via Augmented Lagrangians, Addison-Wesley Publishing Company, Reading, Massachusetts, 1975.Google Scholar
  8. 8.
    Hestenes, M. R.,Multiplier and Gradient Methods, Journal of Optimization Theory and Applications, Vol. 4, pp. 303–320, 1969.Google Scholar
  9. 9.
    Powell, M. J. D.,A Method for Nonlinear Constraints in Minimization Problems, Optimization, Edited by R. Fletcher, Academic Press, New York, New York, 1969.Google Scholar
  10. 10.
    O'Doherty, R. J., andPierson, B. L.,A Numerical Study of Augmented Penalty Function Algorithms for Terminally Constrained Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 14, No. 4, 1974.Google Scholar
  11. 11.
    Tripathi, S. S., andNarendra, K. S.,Constrained Optimization Problems Using Multiplier Methods, Journal of Optimization Theory and Applications, Vol. 9, No. 1, 1972.Google Scholar
  12. 12.
    Miele, A., Gragg, E. E., Iyer, R. R., andLevy, A. V.,Use of the Augmented Penalty Function in Mathematical Programming Problems, Part 1, Journal of Optimization Theory and Applications, Vol. 8, pp. 115–130, 1971.Google Scholar
  13. 13.
    Connor, M. A., andVlach, M.,A New Augmented Penalty Function Technique for Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 21, No. 1, 1977.Google Scholar
  14. 14.
    Miele, A., Moseley, P. E., andCragg, E. E.,A Modification of the Method of Multipliers for Mathematical Programming Problems, Techniques of Optimization, Edited by A. V. Balakrishnan, Academic Press, New York, New York, 1972.Google Scholar
  15. 15.
    Rupp, R. D.,A Method for Solving a Quadratic Optimal Control Problem, Journal of Optimization Theory and Applications, Vol. 9, pp. 238–250, 1972.Google Scholar
  16. 16.
    Rupp, R. D.,A Nonlinear Optimal Control Minimization Technique, Transactions of the American Mathematical Society, Vol. 178, pp. 357–381, 1973.Google Scholar
  17. 17.
    Rupp, R. D.,Convergence and Duality for the Multiplier and Penalty Methods, Journal of Optimization Theory and Applications, Vol. 15, No. 1, 1975.Google Scholar
  18. 18.
    Rockafellar, R. T.,Augmented Lagrange Multiplier Functions and Duality in Nonconvex Programming, SIAM Journal on Control, Vol. 12, pp. 268–285, 1974.Google Scholar
  19. 19.
    Leitmann, G., Editor,Topics in Optimization, Academic Press, New York, New York, 1967.Google Scholar
  20. 20.
    Rockafellar, R. T.,The Multiplier Method of Hestenes and Powell Applied to Convex Programming, Journal of Optimization Theory and Applications, Vol. 12, pp. 555–562, 1973.Google Scholar
  21. 21.
    Tripathi, S. S., andNarendra, K. S.,Constrained Optimization Problems Using Multiplier Methods, Journal of Optimization Theory and Applications, Vol. 9, pp. 59–70, 1972.Google Scholar
  22. 22.
    Ibiejugba, M. A. andOnumanyi, P.,On a Control Operator and Some of Its Applications, Journal of Mathematical Analysis and Applications, Vol. 103, pp. 31–47, 1984.Google Scholar
  23. 23.
    Di Pillo, G., Grippo, L., andLampariello, F.,The Multiplier Method for Optimal Control Problems, Conference on Optimization Engineering and Economics, Naples, Italy, 1974.Google Scholar
  24. 24.
    Miele, A., Cragg, E. E., andLevy, A. V.,Use of the Augmented Penalty Function in Mathematical Programming Problems, Part 2, Journal of Optimization Theory and Applications, Vol. 8, pp. 131–153, 1971.Google Scholar
  25. 25.
    Bertsekas, D. P.,Convergence Rate of the Penalty and Multiplier Methods, Proceedings of the IEEE Conference on Decision and Control, San Diego, California, pp. 260–264, 1973.Google Scholar
  26. 26.
    Miele, A., Moseley, P. E., Levy, A. V., andCoggins, G. M.,On the Method of Multipliers for Mathematical Programming Problems, Journal of Optimization Theory and Applications, Vol. 10, pp. 1–33, 1972.Google Scholar
  27. 27.
    Bertsekas, D. P.,Combined Primal-Dual and Penalty Methods for Constrained Minimization, SIAM Journal on Control, Vol. 13, pp. 521–544, 1975.Google Scholar
  28. 28.
    Rupp, R. D.,Approximation of the Classical Isoperimetric Problem, Journal of Optimization Theory and Applications, Vol. 9, pp. 251–264, 1972.Google Scholar
  29. 29.
    Ibiejugba, M. A., andAbiola, B.,On Application of a Control Operator to Speedy Convergence Rate of Extended Conjugate Gradient Algorithms, Proceedings of the AMSE International Bermuda Symposium on Modelling and Simulation, Hamilton, Bermuda, pp. 32–39, 1983.Google Scholar
  30. 30.
    Ibiejugba, M. A., andAdeboye, K. R.,On the Convergence of a Diffusion Equation, Advances in Modelling and Simulation, Vol. 2, pp. 47–56, 1984.Google Scholar
  31. 31.
    Ibiejugba, M. A.,The Ingenuity of the Method of Multipliers in Solving Optimization Problems, AMSE Review, Vol. 1, pp. 11–22, 1985.Google Scholar
  32. 32.
    Ibiejugba, M. A., andOyatoye, E. O.,Convergence of Optimal Economic Growth in an Aggregative Closed Economy, Proceedings of the International Conference on Numerical Analysis and Its Applications, University of Benin, Benin City, Nigeria, 1983 (to appear).Google Scholar

Copyright information

© Plenum Publishing Corporation 1985

Authors and Affiliations

  • M. A. Ibiejugba
    • 1
  1. 1.Department of Mathematics, Faculty of ScienceUniversity of IlorinKwara StateNigeria

Personalised recommendations