Skip to main content
SpringerLink
Log in

A gradient projection-multiplier method for nonlinear programming

  • Contributed Papers
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper describes a gradient projection-multiplier method for solving the general nonlinear programming problem. The algorithm poses a sequence of unconstrained optimization problems which are solved using a new projection-like formula to define the search directions. The unconstrained minimization of the augmented objective function determines points where the gradient of the Lagrangian function is zero. Points satisfying the constraints are located by applying an unconstrained algorithm to a penalty function. New estimates of the Lagrange multipliers and basis constraints are made at points satisfying either a Lagrangian condition or a constraint satisfaction condition. The penalty weight is increased only to prevent cycling. The numerical effectiveness of the algorithm is demonstrated on a set of test problems.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Betts, J. T.,An Accelerated Multiplier Method for Nonlinear Programming, Journal of Optimization Theory and Applications, Vol. 21, No. 2, 1977.

  2. Powell, M. J. D.,A Method for Nonlinear Constraints in Minimization Problems, Optimization, Edited by R. Fletcher, Academic Press, London, England, p. 283, 1969.

    Google Scholar 

  3. Hestenes, M. R.,Multiplier and Gradient Methods, Journal of Optimization Theory and Applications, Vol. 4, No. 5, 1969.

  4. Fletcher, R.,Minimizing General Functions Subject to Linear Constraints, Numerical Methods for Nonlinear Optimization, Edited by F. A. Lootsma, Academic Press, London, England, 1972.

    Google Scholar 

  5. Murtagh, B. A., andSargent, R. W. H.,A Constrained Minimization Method with Quadratic Convergence, Optimization, Edited by R. Fletcher, Academic Press, London, England, p. 215, 1969.

    Google Scholar 

  6. Fletcher, R., andMcCann, A. P.,Acceleration Techniques for Nonlinear Programming, Optimization, Edited by R. Fletcher, Academic Press, London, England, p. 203, 1969.

    Google Scholar 

  7. Biggs, M. C.,Constrained Minimization Using Recursive Equality Quadratic Programming, Numerical Methods for Nonlinear Optimization, Edited by F. A. Lootsma, Academic Press, London, England, p. 411, 1972.

    Google Scholar 

  8. Murray, W.,An Algorithm for Constrained Minimization, Optimization, Edited by R. Fletcher, Academic Press, London, England, p. 247, 1969.

    Google Scholar 

  9. Betts, J. T.,An Improved Penalty Function Method for Solving Constrained Parameter Optimization Problems, Journal of Optimization Theory and Applications, Vol. 16, Nos. 1–2, 1975.

  10. Betts, J. T.,Solving the Nonlinear Least Square Problem: Application of a General Method, Journal of Optimization Theory and Applications, Vol. 18, No. 4, 1976.

  11. Betts, J. T.,An Accelerated Multiplier Method for Nonlinear Programming, The Aerospace Corporation, El Segundo, California, Report No. TR-0075(5901-03)-5, 1974.

    Google Scholar 

  12. Fiacco, A. V., andMcCormick, G. P.,Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley and Sons, New York, New York, 1968.

    Google Scholar 

  13. Hanson, R. J., andLawson, C. L. Extensions and Applications of the Householder Algorithm for Solving Linear Least Squares Problems, Mathematics of Computation, Vol. 23, No. 108, 1969.

  14. Betts, J. T., andHemenover, A. D.,Optimal Three-Burn Transfer, AIAA Journal, Vol. 15, No. 6, 1977.

  15. Fletcher, R., andPowell, M. J. D.,A Rapidly Convergent Descent Method for Minimization, Computer Journal, Vol. 6, No. 2, 1963.

  16. Rosenbrock, H. H.,An Automatic Method for Finding the Greatest or Least Value of a Function, Computer Journal, Vol. 3, No. 175, 1960.

  17. Leon, A.,A Comparison Among Eight Known Optimizing Procedures, Recent Advances in Optimization Techniques, Edited by A. Lavi and T. P. Vogl, John Wiley and Sons, New York, New York, pp. 23–42,

  18. Beale, E. M. L.,On an Iterative Method for Finding a Local Minimum of a Function of More Than One Variable, Princeton University, Statistical Techniques Research Group, Technical Report No. 25, 1958.

  19. Wampler, R. H.,An Evaluation of Linear Least Squares Computer Programs, Journal of Research of National Bureau of Standards, Vol. 73B, No. 2, 1969.

  20. Colville, A. R.,A Comparative Study on Nonlinear Programming Codes, IBM, New York Scientific Center, Report No. 320–2949, 1968.

  21. Himmelblau, D. M.,Applied Nonlinear Programming, McGraw-Hill Book Company, New York, New York, 1972.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Guidance and Control Division, The Aerospace Corporation, El Segundo, California

    J. T. Betts (Member of the Technical Staff)

Authors
  1. J. T. Betts
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by M. R. Hestenes

The author gratefully acknowledges the helpful suggestions of W. H. Ailor, J. L. Searcy, and D. A. Schermerhorn during the preparation of this paper. The author would also like to thank D. M. Himmelblau for supplying a number of interesting test problems.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Betts, J.T. A gradient projection-multiplier method for nonlinear programming. J Optim Theory Appl 24, 523–548 (1978). https://doi.org/10.1007/BF00935298

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00935298

Key Words

Search

Navigation