Journal of Optimization Theory and Applications

, Volume 12, Issue 6, pp 555–562 | Cite as

The multiplier method of Hestenes and Powell applied to convex programming

  • R. T. Rockafellar
Article

Abstract

For nonlinear programming problems with equality constraints, Hestenes and Powell have independently proposed a dual method of solution in which squares of the constraint functions are added as penalties to the Lagrangian, and a certain simple rule is used for updating the Lagrange multipliers after each cycle. Powell has essentially shown that the rate of convergence is linear if one starts with a sufficiently high penalty factor and sufficiently near to a local solution satisfying the usual second-order sufficient conditions for optimality. This paper furnishes the corresponding method for inequality-constrained problems. Global convergence to an optimal solution is established in the convex case for an arbitrary penalty factor and without the requirement that an exact minimum be calculated at each cycle. Furthermore, the Lagrange multipliers are shown to converge, even though the optimal multipliers may not be unique.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Hestenes, M. R.,Multiplier and Gradient Methods, Computing Methods in Optimization Problems—2, Edited by L. A. Zadeh, L. W. Neustadt, and A. V. Balakrishnan, Academic Press, New York, New York, 1969.Google Scholar
  2. 2.
    Hestenes, M. R.,Multiplier and Gradient Methods, Journal of Optimization Theory and Applications, Vol. 4, pp. 303–320, 1969.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Powell, M. J. D.,A Method for Nonlinear Constraints in Minimization Problems, Optimization, Edited by R. Fletcher, Academic Press, New York, New York, 1972.Google Scholar
  4. 4.
    Miele, A., Cragg, E. E., Iver, 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.MathSciNetCrossRefGoogle Scholar
  5. 5.
    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.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    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
  7. 7.
    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.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Fletcher, R.,A Class of Methods for Nonlinear Programming with Termination and Convergence Properties, Integer and Nonlinear Programming, Edited by J. Abadie, North-Holland Publishing Company, Amsterdam, Holland, 1970.Google Scholar
  9. 9.
    Fletcher, R., andLill, S. A.,A Class of Methods for Nonlinear Programming, II: Computational Experience, Nonlinear Programming, Edited by J. B. Rosen, O. L. Mangasarian, and K. Ritter, Academic Press, New York, New York, 1971.Google Scholar
  10. 10.
    Arrow, K. J., andSolow, R. M.,Gradient Methods for Constrained Maxima, with Weakened Assumptions, Studies in Linear and Nonlinear Programming, Edited by K. Arrow, L. Hurwicz, and H. Uzawa, Stanford University Press, Stanford, California, 1958.Google Scholar
  11. 11.
    Rockafellar, R. T.,A Dual Approach to Solving Nonlinear Programming Problems by Unconstrained Optimization, Mathematical Programming (to appear).Google Scholar
  12. 12.
    Arrow, K. J., Gould, F. J., andHowe, S. M.,A General Saddle Point Result for Constrained Optimization, Mathematical Programming (to appear).Google Scholar
  13. 13.
    Moreau, J. J.,Proximité et Dualité dans un Espace Hilbertien, Bulletin de la Societé Mathématique de France, Vol. 93, pp. 273–279, 1965.MathSciNetMATHGoogle Scholar
  14. 14.
    Fan, K., Glicksberg, I., andHoffman, A. J.,Systems of Inequalities Involving Convex Functions, Proceedings of the American Mathematical Society, Vol. 8, pp. 617–622, 1957.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.MATHGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1973

Authors and Affiliations

  • R. T. Rockafellar
    • 1
  1. 1.Department of MathematicsUniversity of WashingtonSeattle

Personalised recommendations