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On the implementation of reduced gradient methods

  • H. Mukai
  • E. Polak
Mathematical Programming
Part of the Lecture Notes in Computer Science book series (LNCS, volume 41)

Abstract

Until now, the implementation of reduced gradient methods had to be improvised empirically, since procedures for the truncation of the inner iterations, in the feasibility restoration stage, have not been analyzed with respect to convergence of the overall algorithm. This paper presents an implementation of one reduced gradient method. While retaining all the attractive features of the classical reduced gradient methods, this implementation incorporates, explicitly, efficient procedures for truncating the inner iterations to a finite number. In the paper, we present the properties of the restoration subalgorithm and we prove the convergence of the new algorithm under fairly general assumptions.

References

  1. [1]
    J. Abadie and J. Carpentier, Generalization of the Wolfe Reduced Gradient Method for the Case of Nonlinear Constraints," in Optimization, ed. by R. Fletcher, Academic Press, 1969.Google Scholar
  2. [2]
    J. Abadie and J. Guigou, "Numerical Experiments with the GRG Method," in Integer and Nonlinear Programming, ed. by J. Abadie, North-Holland Pub. Co., Amsterdam, 1970.Google Scholar
  3. [3]
    L. Armijo, "Minimization of Functions having Continuous Partial Derivatives," Pacific J. Math., vol. 16, pp. 1–3, 1966.Google Scholar
  4. [4]
    M. D. Canon, C. D. Cullum, Jr. and E. Polak, Theory of Optimal Control and Mathematical Programming, McGraw-Hill, 1970.Google Scholar
  5. [5]
    D. Babay and D. G. Luenberger, "Efficiently Converging Minimization Methods Based on the Reduced Gradient," SIAM Journal on Control, in press.Google Scholar
  6. [6]
    D. G. Luenberger, Introduction to Linear and Nonlinear Programming, Addison-Wesley Pub. Co., 1973.Google Scholar
  7. [7]
    H. Mukai and E. Polak, "On the Use of Approximations in Algorithms for Optimization Problems with Equality and Inequality Constraints," Electronics Research Laboratory Memorandum No. 489, University of California, Berkeley, November 1974.Google Scholar
  8. [8]
    E. Polak, Computational Methods in Optimization, Academic Press, 1971.Google Scholar
  9. [9]
    S. M. Robinson, "Extension of Newton's Method to Mixed Systems of Nonlinear Equations and Inequalities," Tech. Sum. Rept. no. 1161, Mathematical Research Center, University of Wisconsin, 1971.Google Scholar
  10. [10]
    P. Wolfe, "Methods for Nonlinear Constraints," in Nonlinear Programming, ed. by J. Abadie, North-Holland Pub. Co., Amsterdam, 1967.Google Scholar
  11. [11]
    W. I. Zangwill, Nonlinear Programming: A Unified Approach, Prentice-Hall, Englewood Cliffs, New Jersey, 1969.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1976

Authors and Affiliations

  • H. Mukai
    • 1
  • E. Polak
    • 1
  1. 1.Department of Electrical Engineering and Computer Sciences and the Electronics Research LaboratoryUniversity of CaliforniaBerkeley

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