Journal of Optimization Theory and Applications

, Volume 12, Issue 3, pp 233–241 | Cite as

Global convergence for Newton methods in mathematical programming

  • J. W. Daniel


In constrained optimization problems in mathematical programming, one wants to minimize a functionalf(x) over a given setC. If, at an approximate solutionx n , one replacesf(x) by its Taylor series expansion through quadratic terms atx n and denotes byx n+1 the minimizing point for this overC, one has a direct analogue of Newton's method. The local convergence of this has been previously analyzed; here, we give global convergence results for this and the similar algorithm in which the constraint setC is also linearized at each step.


Newton Method Global Convergence Local Convergence Constraint Qualification Quadratic Convergence 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ortega, J., andRheinboldt, W.,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.MATHGoogle Scholar
  2. 2.
    Elkin, R.,Convergence Theorems for Gauss-Seidel and Other Minimization Algorithms, University of Maryland, Computer Science Report No. 68–59, 1968.Google Scholar
  3. 3.
    Daniel, J.,The Approximate Minimization of Functionals, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.MATHGoogle Scholar
  4. 4.
    Levitin, E., andPoljak, B.,Constrained Minimization Methods, USSR Computational Mathematics and Mathematical Physics, Vol. 6, pp. 1–50, 1968.CrossRefGoogle Scholar
  5. 5.
    Auslender, A.,Méthodes Numériques pour la Résolution des Problèmes d'Optimisation avec Contraintes, University of Grenoble, Ph.D. Thesis, 1969.Google Scholar
  6. 6.
    Auslender, A.,Méthode du Second Ordre dans les Problèmes d'Optimisation avec Contraintes, Revue Française d'Informatique et de Recherche Opérationnelle, Vol. 2, pp. 27–42, 1969.MathSciNetGoogle Scholar
  7. 7.
    Rosen, J. B.,Iterative Solution of Nonlinear Optimal Control Problems, SIAM Journal on Control, Vol. 4, pp. 223–244, 1966.MATHCrossRefGoogle Scholar
  8. 8.
    Meyer, R.,The Solution of Non-Convex Optimization Problems by Iterative Convex Programming, University of Wisconsin, Ph.D. Thesis, 1968.Google Scholar
  9. 9.
    Meyer, R.,The Validity of a Family of Optimization Methods, SIAM Journal on Control, Vol. 8, pp. 41–54, 1970.MATHCrossRefGoogle Scholar
  10. 10.
    Robinson, S.,Private Communication, 1972.Google Scholar
  11. 11.
    Mangasarian, O.,On a Newton Method for Reverse-Convex Programming, University of Wisconsin, Computer Sciences TR No. 146, 1972.Google Scholar
  12. 12.
    Armijo, L.,Minimization of Functionals Having Lipschitz Continuous First Partial Derivatives, Pacific Journal of Mathematics, Vol. 16, pp. 1–3, 1966.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Daniel, J.,Convergent Step-Sizes for Gradient-Like Feasible Direction Algorithms for Constrained Optimization, Nonlinear Programming, Edited by J. B. Rosenet al., Academic Press, New York, New York, 1970.Google Scholar
  14. 14.
    Daniel, J.,Convergent Step-Sizes for Curvilinear-Path Methods of Minimization, Techniques of Optimization, Edited by A. V. Balakrishnan, Academic Press, New York, New York, 1972.Google Scholar
  15. 15.
    Mangasarian, O.,Nonlinear Programming, McGraw-Hill Book Company, New York, New York, 1969.MATHGoogle Scholar
  16. 16.
    Arrow, K. J., Hurwicz, L., andUzawa, H.,Constraint Qualifications in Maximization Problems, Naval Research and Logistics Quarterly, Vol. 8, pp. 175–191, 1961.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Ostrowski, A.,Contributions to the Theory of the Method of Steepest Descent, I, University of Wisconsin, Mathematics Research Center Report No. 615, 1966.Google Scholar
  18. 18.
    Ostrowski, A.,Solution of Equations and Systems of Equations, Academic Press, New York, New York, 1966.MATHGoogle Scholar
  19. 19.
    Daniel, J.,Stability of the Solution and of the Constraint Set in Quadratic Programming, University of Texas, Center for Numerical Analysis, Report No. 58, 1972.Google Scholar

Copyright information

© Plenum Publishing Corporation 1973

Authors and Affiliations

  • J. W. Daniel
    • 1
    • 2
  1. 1.Departments of Mathematics and of Computer SciencesThe University of Texas at AustinAustin
  2. 2.Center for Numerical AnalysisThe University of Texas at AustinAustin

Personalised recommendations