Zusammenfassung
Von einem allgemeinen Standpunkt aus werden verschiedene Schrittweitenfunktionen diskutiert und ihr Einfluß auf die Konvergenz von Verfahren der unrestringierten Optimierung betrachtet. Es werden effiziente Schrittweitenfunktionen definiert und gezeigt, daß die bekannten Schrittweitenalgorithmen effizient sind. Schließlich werden notwendige und hinreichende Konvergenzkriterien für Abstiegsverfahren angegeben und auf Verfahren der konjugierten Gradienten angewendet.
Abstract
In the present paper we discuss several steplength procedures from a general point of view. We consider their influence on the convergence of algorithms for the numerical treatment of optimization problems without constraints. We define efficient step-size functions and show that well known steplength procedures are efficient. Necessary and sufficient conditions for convergence of descent methods with efficient step-size functions and applications to conjugate gradient methods are given.
This is a preview of subscription content, log in via an institution to check access.
Literatur
Daniel, J. W.: The approximate minimization of functionals. Englewood Cliffs, N.J.: Prentice-Hall 1971.
Danilin, Y. M., Psenichnyi, B. N.: On methods of minimization with accelerated convergence. USSR, Comp. Math. a. Math. Phys.10, 4–19 (1972).
Goldstein, A. A., Price, I. F.: An effective algorithm for minimization. Num. Math.10, 184–189 (1967).
Lenard, M. L.: Practical convergence conditions for unconstrained optimization. Math. Prog.4, 309–323 (1973).
Ortega, J. M., Rheinboldt, W. C.: Iterative solution of nonlinear equations in several variables. New York: Academic Press 1970.
Ortega, J. M., Rheinboldt, W. C.: A general convergence result for unconstrained minimization methods. SIAM J. Num. Anal.9, 40–43 (1972).
Polak, E.: Computational methods in optimization. New York: Academic Press 1971.
Ritter, K.: A superlinearly convergent method for unconstrained minimization problems, in: Nonlinear programming (Rosen, I. B., Mangasarian, O. L., Ritter, K., eds.). New York: Academic Press 1970.
Wolfe, Ph.: Convergence conditions for ascent methods. SIAM Rev.11, 226–235 (1969).
Wolfe, Ph.: Convergence theory in nonlinear programming, in: Integer and nonlinear programming (Abadie, J., ed.). Amsterdam: North-Holland 1970.
Zoutendijk, G.: Some algorithms based on the principle of feasible directions, in: Integer and nonlinear programming (Abadie, J., ed.). Amsterdam: North-Holland 1970.
Fletcher, R., Reeves, C. M.: Function minimization by conjugate gradients. Comp. J.7, 149–154 (1964).
Polak, E., Ribière, G.: Note sur la convergence de méthodes de direction conjugées. RIRO3, 35–44 (1969).
Polyak, B. T.: Conjugate gradient methods in extremum problems. USSR Comp. Math. a. Math. Phys.9, 94–112 (1969).
Mc Cormick, G. P., Ritter, K.: Alternative proofs of the convergence of the conjugate method. JOTA13, 497–518 (1974).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Warth, W., Werner, J. Effiziente Schrittweitenfunktionen bei unrestringierten Optimierungsaufgaben. Computing 19, 59–72 (1977). https://doi.org/10.1007/BF02260741
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02260741