Abstract
This paper gives a general safeguarded bracketing technique for minimizing a function of a single variable. In certain cases the technique guarantees convergence to a stationary point and, when combined with sequential polynomial and/or polyhedral fitting algorithms, preserves rapid convergence. Each bracket has an interior point whose function value does not exceed those of the two bracket endpoints. The safeguarding technique consists of replacing the fitting algorithm's iterate candidate by a close point whose distance from the three bracket points exceeds a positive multiple of the square of the bracket length. It is shown that a given safeguarded quadratic fitting algorithm converges in a certain better than linear manner with respect to the bracket endpoints for a strongly convex twice continuously differentiable function.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.P. Brent,Algorithms for Minimization Without Derivatives (Prentice-Hall, Englewood Cliffs, NJ, 1973).
P.E. Gill and W. Murray, “Safeguarded steplength algorithms for optimization using descent methods,” National Physical Laboratory Report NAC 37 (Teddington, 1974).
C. Lemarechal and R. Mifflin, “Global and superlinear convergence of an algorithm for one-dimensional minimization of convex functions,”Mathematical Programming 24 (1982) 241–256.
R. Mifflin, “Stationarity and superlinear convergence of an algorithm for univariate locally Lipschitz constrained minimization,”Mathematical Programming 28 (1984) 50–71.
R. Mifflin, “Better than linear convergence and safeguarding in nonsmooth minimization,” in: P. Thoft-Christensen, ed.,System Modelling and Optimization (Springer-Verlag, Berlin, 1984) 321–330.
R. Mifflin and J.-J. Strodiot, “A rapidly convergent five point algorithm for univariate minimization,” Report 85/12, Department of Mathematics, Facultés Universitaires de Namur (Namur, 1985).
S.M. Robinson, “Quadratic interpolation is risky,”SIAM Journal on Numerical Analysis 16 (1979) 377–379.
R.T. Rockafellar,The Theory of Subgradients and its Application to Problems of Optimization. Convex and Nonconvex Functions (Heldermann Verlag, Berlin, 1981).
M.A. Wolfe,Numerical Methods for Unconstrained Optimization (Van Nostrand Reinhold Company, New York, 1978).
Author information
Authors and Affiliations
Additional information
Research sponsored by the Institut National de Recherche en Informatique et en Automatique, Rocquencourt, France, and by the Air Force Office of Scientific Research, Air Force System Command, USAF, under Grant Number AFOSR-83-0210. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.
Research sponsored, in part, by the Institut National de Recherche en Informatique et en Automatique, Rocquencourt, France.
Rights and permissions
About this article
Cite this article
Mifflin, R., Strodiot, J.J. A bracketing technique to ensure desirable convergence in univariate minimization. Mathematical Programming 43, 117–130 (1989). https://doi.org/10.1007/BF01582285
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01582285