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**Multiplier and gradient methods.**
*(English)*
Zbl 0174.20705

Summary: The main purpose of this paper is to suggest a method for finding the minimum of a function \(f(x)\) subject to the constraint \(g(x)=0\). The method consists of replacing \(f\) by \(F=f+\lambda g+ \tfrac12 cg^2\), where \(c\) is a suitably large constant, and computing the appropriate value of the Lagrange multiplier. Only the simplest algorithm is presented.

The remaining part of the paper is devoted to a survey of known methods for finding unconstrained minima, with special emphasis on the various gradient techniques that are available. This includes Newton’s method and the method of conjugate gradients.

The remaining part of the paper is devoted to a survey of known methods for finding unconstrained minima, with special emphasis on the various gradient techniques that are available. This includes Newton’s method and the method of conjugate gradients.

### Keywords:

minimum of a function subject to constraint; Lagrange multiplier; survey of known methods; finding unconstrained minima; gradient techniques; Newton’s method; method of conjugate gradients
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\textit{M. R. Hestenes}, J. Optim. Theory Appl. 4, 303--320 (1969; Zbl 0174.20705)

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### References:

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[2] | Mengel, A. S.,Optimum Trajectories, The RAND Corporation, Report No. P-199, 1951. |

[3] | Hestenes, M. R.,Numerical Methods for Obtaining Solutions of Fixed-Endpoint Problems in the Calculus of Variations, The RAND Corporation, Research Memorandum No. RM-102, 1949. |

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[9] | Hestenes, M. R.,The Conjugate Gradient Method for Solving Linear Systems, Proceedings of the Sixth Symposium in Applied Mathematics, Edited by J. H. Curtiss, American Mathematical Society, Providence, Rhode Island, 1956. · Zbl 0072.14102 |

[10] | Hayes, R. M.,Iterative Methods for Solving Linear Problems in Hilbert Space, Contributions to the Solutions of Systems of Linear Equations and the Determinations of Eigenvalues, Edited by O. Tausky, National Bureau of Standards, Applied Mathematics Series, US Government Printing Office, Washington, D.C., 1954. · Zbl 0058.10703 |

[11] | Fletcher, R., andPowell, M. J. D.,A Rapidly Convergent Descent Method for Minimization, Computer Journal, Vol. 6, No. 2, 1964. · Zbl 0132.11603 |

[12] | Myers, G. E.,Properties of the Conjugate-Gradient and Davidon Methods, Journal of Optimization Theory and Applications, Vol. 2, No. 4, 1968. · Zbl 0207.17302 |

[13] | Horwitz, L. B., andSarachick, P. E.,Davidon’s Method in Hilbert Space, SIAM Journal on Applied Mathematics, Vol. 16, No. 4, 1968. · Zbl 0159.43803 |

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