Generation of conjugate directions for unconstrained minimization without derivatives

Author:
Larry Nazareth

Journal:
Math. Comp. **30** (1976), 115-131

MSC:
Primary 65K05

DOI:
https://doi.org/10.1090/S0025-5718-1976-0398100-2

MathSciNet review:
0398100

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We analyze a technique for unconstrained minimization without derivatives. This stems from two theorems proved by M. J. D. Powell. A particular version, which we consider in detail, is related to the Jacobi process for finding the eigensystem of a symmetric matrix, and the two processes, although different, help to illuminate one another. We study convergence of the search directions to mutual conjugacy, cases when cycling occurs and identify a broad class of 'cyclic patterns' for which convergence to mutual conjugacy is proven.

**[1]**M. J. D. POWELL, "An efficient method for finding the minimum of a function of several variables without calculating derivatives,"*Comput. J.*, v. 7, 1964, pp. 155-162. MR**32**#4828. MR**0187376 (32:4828)****[2]**W. I. ZANGWILL, "Minimizing a function without calculating derivatives,"*Comput. J.*, v. 10, 1967, pp. 293-296. MR**38**#2930. MR**0234614 (38:2930)****[3]**R. P. BRENT,*Algorithms for Minimization Without Derivatives*, Prentice-Hall Ser. in Automatic Computation, Prentice-Hall, Englewood Cliffs, N. J., 1973. MR**49**#4251. MR**0339493 (49:4251)****[4]**N. W. BRODLIE,*A New Method for Unconstrained Minimization without Evaluating Derivatives*, IBM Report UKSC-0019-0019, IBM, Peterlee, United Kingdom, 1972.**[5]**M. J. D. POWELL,*Unconstrained Minimization Algorithms Without Computation of Derivatives*, UK. A. E. A. Harwell Report HL72/1713, 1972.**[6]**J. L. NAZARETH, Part I:*Unified Approach to Unconstrained Minimization*. Part II:*Generation of Conjugate Directions for Unconstrained Minimization Without Derivatives*, Dept. of Computer Science, Univ. of California, Berkeley, Rep. No. 23, 1973.**[7]**ELDON R. HANSEN, "On cyclic Jacobi methods,"*J. Soc. Indust. Appl. Math.*, v. 11, 1963, pp. 448-459. MR**27**#6381. MR**0156458 (27:6381)****[8]**J. L. NAZARETH,*On the Convergence of the Cyclic Jacobi Process*, Argonne National Lab. Tech. Memo, ANL-AMD, 1974.**[9]**J. H. WILKINSON, "Note on the quadratic convergence of the cyclic Jacobi process,"*Numer. Math.*, v. 4, 1962, pp. 296-300. MR**26**#4473. MR**0146954 (26:4473)****[10]**J. H. WILKINSON,*The Algebraic Eigenvalue Problem*, Clarendon Press, Oxford, 1965. MR**32**#1894. MR**0184422 (32:1894)****[11]**G. E. FORSYTHE & P. HENRICI, "The cyclic Jacobi method for computing the principal values of a complex matrix,"*Trans. Amer. Math. Soc.*, v. 94, 1960, pp. 1-23. MR**22**#710. MR**0109825 (22:710)****[12]**PETER HENRICI, "On the speed of convergence of cyclic and quasicyclic Jacobi methods for computing eigenvalues of Hermitian matrices,"*J. Soc. Indust. Appl. Math.*, v. 6, 1958, pp. 144-162. MR**20**#2084. MR**0095582 (20:2084)****[13]**H. P. M. VAN KEMPEN, "On the convergence of the classical Jacobi method for real symmetric matrices with non-distinct eigenvalues,"*Numer. Math.*, v. 9, 1966, pp. 11-18. MR**34**#2163. MR**0202290 (34:2163)****[14]**J. N. FRANKLIN,*Matrix Theory*, Prentice-Hall, Englewood Cliffs, N. J., 1968. MR**38**#5798. MR**0237517 (38:5798)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65K05

Retrieve articles in all journals with MSC: 65K05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1976-0398100-2

Article copyright:
© Copyright 1976
American Mathematical Society