A Comparison of Some Recent Iterative Methods for the Numerical Solution of Nonlinear Programs

  • E. J. Beltrami
Part of the Lecture Notes in Operations Research and Mathematical Economics book series (LNE, volume 14)


In this paper we wish to survey some recent algorithms for constrained minimization on Rn and, in particular, for nonlinear programs. Our emphasis will be on the numerical difficulties associated with these methods, along with some discussion of that can be done about overcoming such computational problems.


Steep Descent Conjugate Gradient Method Penalty Term Nonlinear Program Gradient Projection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. (1).
    Davidon, W.C.: Variable Metric Methods for Minimization. A.E.C.Research Report ANL-5990 (1959).Google Scholar
  2. (2).
    Fletcher, R. and M. J. Powell: A Rapidly Convergent Descent Method for Minimization. Computer J. 6 (1963), 163–168.CrossRefGoogle Scholar
  3. (3).
    Pearson, J.D.: On Variable Metric Methods of Minimization. RAC Report RAC-TP-302, 1968.Google Scholar
  4. (4).
    Broyden, C.G.: Quasi-Newton Methods and their Application to Function Minimization. Math. Comput. 21 (1967), 368381.Google Scholar
  5. (5).
    ).Zeleznik, F.J.: Quasi-Newton Methods for Nonlinear Equations. J.A.C.M. 15, (1968), 265–271.Google Scholar
  6. (6).
    Fletcher, R. and C.M. Reeves: Function Minimization by Conjugate Gradients. Computer J. 7 (1964), 149–153•Google Scholar
  7. (7).
    Hestenes, M.R. and E. Stiefes: Method of Conjugate Gradients for Solving Linear Systems. Report 1659, N.B.S. (1952).Google Scholar
  8. (8).
    Myers, G.: Properties of the Conjugate Gradient and Davi-don Methods. J. Optimiz. Th. amp; Applic. 2 (1968), 209–219.CrossRefGoogle Scholar
  9. (9).
    Powell, M.J.D.: An Efficient Method for Finding the Minimum of a Function without Calculating Derivatives. Computer J. 7 (1964), 155–162.CrossRefGoogle Scholar
  10. (10).
    Zangwill, W.I.: Minimizing a Function without Calculating Derivatives. Computer J. 10 (1967), 293–296.CrossRefGoogle Scholar
  11. (11).
    Goldfarb, D. and L. Lapidus: A Conjugate Gradient Method for Nonlinear Programming Problems with Linear Constraints. Indust. amp; Eng. Chem. Fund. 7 (1968), 1112–151.Google Scholar
  12. (12).
    Goldfarb, D.: Extension of Davidon’s Variable Metric Method to Maximization under Linear Inequality and Equality Constraints. To appear in SIAM J. 1968.Google Scholar
  13. (13).
    Rosen, J.B.: The Gradient Projection Method for Nonlinear Programming, Part I: Linear Constraints. SIAM J. 8 (1960), 181–217.Google Scholar
  14. (14).
    Courant, R.: Variational Methods for the Solution of Problems of Equilibrium and Vibrations. Bull. AMS 49 (1943), 1–23.CrossRefGoogle Scholar
  15. (15).
    Butler, T. and A. V. Martin: On a Method of Courant for Minimizing Functionals. J. Math. Phy. 111 (1962), 291–299.Google Scholar
  16. (16).
    Beltrami, E. J.: On Infinite Dimensional Convex Programs J. Computer System Sci. 1 (1967), 323–329.CrossRefGoogle Scholar
  17. (17).
    Beltrami, E. J.: A Constructive Proof of the Kuhn-Tucker Multiplier Rule. To appear in J.M.A.A., 1968.Google Scholar
  18. (18).
    Kelley, H. J. and W. Denham, I. Johnson, P. Wheatley: An Accelerated Gradient Method for Parameter Optimization with Nonlinear Constraints. Jour. Astro. Sci. 13 (1966), 166–169.Google Scholar
  19. (19).
    Carroll, C. W.: The Created Response Surface Technique for Optimizing Nonlinear Restrained Systems. Operations Res. 9 (1961), 169–1811.CrossRefGoogle Scholar
  20. (20).
    Fiacco, A. V. and G. McCormick: Programming under Nonlinear Constraints by Unconstrained Minimization. RAC Report RAC-TP-96 (1963).Google Scholar
  21. (21).
    Fiacco, A. V. and G. McCormick: Computational Algorithm for Sequential Unconstrained Minimization Technique for Nonlinear Programming. Management Sci. 10 (19611), 601–617.Google Scholar
  22. (22).
    Fiacco, A. V. and G. McCormick: The Slacked Unconstrained Minimization Technique for Convex Programming. SIAM J. 15, (1967), 505–515.Google Scholar
  23. (23).
    Johnson, I. and G. Myers: One Dimensional Minimization using Search by Golden Section and Cubic Fit Methods. NASA Report 67-FM-172, (1967).Google Scholar
  24. (24).
    Flanagan, P. and P. Vitale, J. Mendelsohn: A Numerical Investigation of Several One-Dimensional Search Procedures in Nonlinear Regression Problems. GAEC Report RE-300, (1967).Google Scholar
  25. (25).
    Bard, Y.: On a Numerical Instability of Davidon-Like Methods. Math. Comput. 22 (1968), 665–666.CrossRefGoogle Scholar
  26. (26).
    Box, M. J.: A Comparison of Several Current Optimization Methods, and the use of Transformations in Constrained Problems. Computer J. 8 (1965), 67–77.CrossRefGoogle Scholar
  27. (27).
    McCormick, G. and J. D. Pearson: Variable Metric Methods and Unconstrained Optimization. Joint Conf. on Optimization, Univ. of Keele, (1968).Google Scholar
  28. (28).
    Kelley, H. J. and G. E. Myers: Conjugate Direction Methods for Parameter Optimization. 18th Congress of I.A.S., Belgrade, (1967).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1969

Authors and Affiliations

  • E. J. Beltrami
    • 1
  1. 1.Department of Applied AnalysisState University of New YorkStony BrookUSA

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