Abstract
In a recent paper McCormick and Ritter consider two classes of algorithms, namely methods of conjugate directions and quasi-Newton methods, for the problem of minimizing a function ofn variablesF(x). They show that the former methods possess ann-step superlinear rate of convergence while the latter are every step superlinear and therefore inherently superior. In this paper a simple and computationally inexpensive modification of a method of conjugate directions is presented. It is shown that the modified method is a quasi-Newton method and is thus every step superlinearly convergent. It is also shown that under certain assumptions on the second derivatives ofF the rate of convergence of the modified method isn-step quadratic.
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References
M.J. Best, “A method to accelerate the rate of convergence of a class of optimization algorithms”, Mathematical Programming 8 (1975) 139–160.
M.J. Best and K. Ritter, “A class of accelerated conjugate direction methods for linearly constrained minimization problems”,Mathematics of Computation 30 (1976) 478–504.
C.G. Broyden, J.E. Dennis and J.J. Moré, “On the local and superlinear convergence of Quasi-Newton methods”,Journal of the Institute of Mathematics and its Applications 12 (1973) 223–245.
A.A. Goldstein,Constructive real analysis (Harper and Row, New York, 1967).
G.P. McCormick and K. Ritter, “Methods of conjugate directions versus quasi-Newton methods”,Mathematical Programming 3 (1) (1972) 101–116.
W.C. Rheinboldt and J.S. Vandergraft, “On the local convergence of update methods”, Tech. Rept. TR255, Computer Science Center, University of Maryland, College Park, MD (1973).
K. Ritter, “Accelerating procedures for methods of conjugate directions”,Computing 14 (1975) 79–105.
J. Stoer, “On the convergence rate of imperfect minimization algorithms in Broyden'sβ-class”,Mathematical Programming 9 (1975) 313–335.
M.M. Vainberg,Variational methods for the study of nonlinear operators (Holden-Day, San Francisco, CA., 1964).
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This work was supported by the National Research Council of Canada under Research Grant A8189.
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Best, M.J. A quasi-Newton method can be obtained from a method of conjugate directions. Mathematical Programming 15, 189–199 (1978). https://doi.org/10.1007/BF01609017
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DOI: https://doi.org/10.1007/BF01609017