Skip to main content
SpringerLink
Log in

Nonlinear leastpth optimization and nonlinear programming

  • Published:
Mathematical Programming Submit manuscript

Abstract

Over the past few years a number of researchers in mathematical programming became very interested in the method of the Augmented Lagrangian to solve the nonlinear programming problem. The main reason being that the Augmented Lagrangian approach overcomes the ill-conditioning problem and the slow convergence of the penalty methods. The purpose of this paper is to present a new method of solving the nonlinear programming problem, which has similar characteristics to the Augmented Lagrangian method. The original nonlinear programming problem is transformed into the minimization of a leastpth objective function which under certain conditions has the same optimum as the original problem. Convergence and rate of convergence of the new method is also proved. Furthermore numerical results are presented which illustrate the usefulness of the new approach to nonlinear programming.

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. R.R. Allan and S.E.J. Johnsen, “An algorithm for solving nonlinear programming problems subject to nonlinear inequality constraints”,The Computer Journal 13 (1970) 171–177.

    Google Scholar 

  2. J. Asaadi, “A computational comparison of some non-linear programs”,Mathematical Programming 4 (1973) 144.

    Google Scholar 

  3. J.W. Bandler and C. Charalambous, “A new approach to nonlinear programming”, in:Proceedings of the fifth Hawaii international conference on system sciences (University of Hawaii, Honolulu, HI, January 1972).

    Google Scholar 

  4. J.W. Bandler and C. Charalambous, “Nonlinear programming using minimax techniques”,Journal of Optimization Theory and Applications 13 (1974) 607–619.

    Google Scholar 

  5. J.W. Bandler and C. Charalambous, “Practical leastpth optimization of networks”,IEEE Transactions on Microwave Theory and Technology (1972 Symposium Issue) MTT-20 (1972) 834–840.

    Google Scholar 

  6. D.P. Bertsekas, “On penalty and multiplier methods for constrained optimization”, in:Proceedings of the nonlinear programming symposium (University of Wisconsin, Madison, WI, 1974).

    Google Scholar 

  7. M.J. Best and K. Ritter, “A class of accelerated conjugate direction methods for linear constrained minimization problems”, Rept. CORR 74-12, dept. of combinatorics and optimization, Univ. Waterloo, Waterloo, Ont., Canada (1974).

    Google Scholar 

  8. C. Charalambous and J.W. Bandler, “Nonlinear minimax optimization as a sequence of leastpth optimization with finite values ofp”,International Journal of Systems Science 7 (1976) 377–391.

    Google Scholar 

  9. A.R. Conn, “Constrained optimization using nondifferentiable penalty function”,SIAM Journal on Numerical Analysis 10 (1973) 760–784.

    Google Scholar 

  10. A.V. Fiacco and G.P. McCormick,Nonlinear programming: sequential unconstrained minimization techniques (Wiley, New York, 1968).

    Google Scholar 

  11. R. Fletcher, “A new approach to variable metric algorithms”,The Computer Journal 13 (1970) 317–322.

    Google Scholar 

  12. R. Fletcher, “An exact penalty function for nonlinear programming with inequalities”,Mathematical Programming 5 (1973) 129–150.

    Google Scholar 

  13. P.C. Haarhoff and J.D. Buys, “A new method for the optimization of a nonlinear function subject to nonlinear constraints”,The Computer Journal 13 (1970) 178.

    Google Scholar 

  14. M.R. Hestenes, “Multiplier and gradient methods”,Journal of Optimization Theory and Applications 4 (1969) 303.

    Google Scholar 

  15. L.S. Lasdon,Optimization theory for large systems (Macmillan, New York, 1970).

    Google Scholar 

  16. F.A. Lootsma, “A survey of methods for solving constrained optimization problems via unconstrained minimization”, in: F.A. Lootsma, ed.,Numerical methods for nonlinear programming (Academic Press, New York, 1972).

    Google Scholar 

  17. T. Pietrzykowski, “An exact potential method for constrained maxima”,SIAM Journal on Numerical Analysis 6 (1969) 299–304.

    Google Scholar 

  18. V.T. Polyak and N.V. Tret'yakov, “The method of penalty estimates for conditional extremum problems”,Žurnal Vyčhislitel'noi Matematiki i Matematičeskoi Fiziki 13 (1973) 34–36.

    Google Scholar 

  19. M.J.D. Powell, “A method for nonlinear constraints in minimization problems”, in: R. Fletcher, ed.,Optimization (Academic Press, 1969).

  20. J.R. Rice,The approximation of functions (Addison-Wesley, Reading, MA, 1969).

    Google Scholar 

  21. R.T. Rockafellar, “A dual approach to solving nonlinear programming problems by unconstrained optimization”,Mathematical Programming 5 (1973) 354–373.

    Google Scholar 

  22. R.T. Rockafellar, “Augmented Lagrangian multiplier functions and duality in nonconvex programming”,SIAM Journal on Control 12 (1974) 268–285.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Waterloo, Waterloo, Ontario, Canada

    Christakis Charalambous

Authors
  1. Christakis Charalambous
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was supported by the National Research Council of Canada and by the Department of Combinatorics and Optimization of the University of Waterloo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Charalambous, C. Nonlinear leastpth optimization and nonlinear programming. Mathematical Programming 12, 195–225 (1977). https://doi.org/10.1007/BF01593788

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01593788

Keywords

Search

Navigation