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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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A bench mark experiment for minimization algorithms
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by J. N. Lyness PDF
Math. Comp. 33 (1979), 249-264 Request permission


In this paper we suggest a single bench mark problem family for use in evaluating unconstrained minimization algorithms or routines. In essence, this problem consists of measuring, for each algorithm, the rate at which it descends an unlimited helical valley. The periodic nature of the problem allows us to exploit affine scale invariance properties of the algorithm. As a result, the capacity of the algorithm to minimize a wide range of helical valleys of various scales may be summarized by calculating a single valued function ${g_Q}({X_1})$. The measurement of this function is not difficult, and the result provides information of a simple, general character for use in decisions about choice of algorithm.
  • William C. Davidon, Optimally conditioned optimization algorithms without line searches, Math. Programming 9 (1975), no. 1, 1–30. MR 383741, DOI 10.1007/BF01681328
  • C. DAVIDON & L. NAZARETH (1977), DRVOCR-A Fortran Implementation of Davidon’s Optimally Conditioned Method, ANL-AMD Technical Memorandum No. 306. FLETCHER (1972), "Conjugate direction methods," Numerical Methods for Unconstrained Optimization, (W. Murray, Editor), Academic Press, London, pp. 73-86. E. GILL, W. MURRAY, S. M. PICKEN, S. R. GRAHAM & M. H. WRIGHT (1975), Subroutine QNMDER, A Quasi-Newton Algorithm to Find the Unconstrained Minimum of a Function of N Variables When First Derivatives are Available, Technical Memorandum E4/02/0/Fortran/11/75, National Physical Laboratory, Teddington, Middlesex TW11 OLW, England.
  • J. N. Lyness, A bench mark experiment for minimization algorithms, Math. Comp. 33 (1979), no. 145, 249–264. MR 514822, DOI 10.1090/S0025-5718-1979-0514822-7
  • N. LYNESS & C. GREENWELL (1977), A Pilot Scheme for Minimization Software Evaluation, ANL-AMD Technical Memorandum No. 323.
  • M. J. D. Powell, Some global convergence properties of a variable metric algorithm for minimization without exact line searches, Nonlinear programming (Proc. Sympos., New York, 1975) SIAM-AMS Proc., Vol. IX, Amer. Math. Soc., Providence, R.I., 1976, pp. 53–72. MR 0426428
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Additional Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Math. Comp. 33 (1979), 249-264
  • MSC: Primary 65K05; Secondary 90C30
  • DOI:
  • MathSciNet review: 514822