Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A second-order method for unconstrained optimization


This paper presents a quadratically converging algorithm for unconstrained minimization. All the accumulation points that it constructs satisfy second-order necessary conditions of optimality. Thus, it avoids second-order saddle andinflection points, an essential feature for a method to be used in minimizing the modified Lagrangians in multiplier methods.

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


  1. 1.

    Goldstein, A. A.,Constructive Real Analysis, Harper and Row Publishers, New York, New York, 1967.

  2. 2.

    Huang, T. J.,Algorithms for Solving Systems of Equations and Inequalities with Applications in Nonlinear Programming, University of Wisconsin, Computer Science Technical Report No. 191, 1973.

  3. 3.

    Polak, E., andTeodoru, I.,Newton Derived Methods for Nonlinear Equations and Inequalities, Nonlinear Programming 2, Edited by O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, Academic Press, New York, New York, pp. 255–277, 1975.

  4. 4.

    Polak, E.,On the Global Stabilization of Locally Convergent Algorithms for Optimization and Root Finding, Automatica, Vol. 12, pp. 337–342, 1976.

  5. 5.

    Hestenes, M. R.,Multiplier and Gradient Methods, Journal of Optimization Theory and Applications, Vol. 4, pp. 303–320, 1969.

  6. 6.

    Powell, M. J. D.,A Method for Nonlinear Constraints in Minimization Problems, Optimization, Edited by R. Fletcher, Academic Press, New York, New York, pp. 283–298, 1964.

  7. 7.

    McCormick, G.,Second-Order Method for the Linear Constrained Nonlinear Programming Problem, Nonlinear Programming, Edited by J. B. Rosen, O. L. Mangasarian, and K. Ritter, Academic Press, New York, New York, pp. 207–243, 1970.

  8. 8.

    Polak, E.,Computational Methods in Optimization, Academic Press, New York, New York, 1971.

  9. 9.

    Luenberger, D. G.,Introduction to Linear and Nonlinear Programming, Addison-Wesley Publishing Company, Reading, Massachusetts, 1973.

  10. 10.

    Kantorovich, L. V., andAkilov, G. P.,Function Analysis is Normed Spaces, Pergamon Press, New York, New York, 1964.

Download references

Author information

Additional information

The work of the first author was supported by NSF RANN AEN 73-07732-A02 and JSEP Contract No. F44620-71-C-0087; the work of the second author was supported by NSF Grant No. GK-37672 and the ARO Contract No. DAHCO4-730C-0025.

Communicated by D. Q. Mayne

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mukai, H., Polak, E. A second-order method for unconstrained optimization. J Optim Theory Appl 26, 501–513 (1978).

Download citation

Key Words

  • Unconstrained optimization
  • quadratic convergence
  • second-order conditions