Die Konvergenzordnung des Fletcher‐Powell‐Algorithmus

Abstract

Das betrachtete Verfahren gehört zu der Klasse von Minimterungsverfahren, die das Minimum einer quadratischen Funktion von n Argumenten in höchstens n Schritten liefern. Für nichtquadratische konvexe Zielfunktionen wird gezeigt, daß das Verfahren mindestens die Konvergenzordnung n√2 besitzt.

On the relation between quadratic termination and convergence properties of minimization algorithms

“…We illustrate this point too by the above example: In this paper we will show in Section 2 that the condition A(n) is satisfied for the familiar conjugate-gradient-algorithms of Fletcher and Reeves [10], Polak and Ribi6re [19] and Daniel [6], thereby reestablishing Cohen's [5] and Lenard's [13] convergence results. In Section 3 we show that large sub-classes of the class of rank-2 algorithms described by Oren and Luenberger [16,17] have the properties A(n) and B(n) respectively: This will among others reestablish the convergence-results of Burmeister [4] and [20,21] for the fl-class of Broyden [2,3], and will generalize these results to a much broader class of methods which contains almost all rank-2-methods of practical interest. In Section 4 A(n) is shown for the rank-l-algorithms of Pearson and McCormick described in Pearson [18].…”

A Three-Term Conjugate Gradient Algorithm with Quadratic Convergence for Unconstrained Optimization Problems

“…and a modified weak Wolfe-Powell line search technique is presented to study open unconstrained optimization (see [27,28]). If a restart strategy is used, the PRP algorithm is n-step quadratic convergence (see [29][30][31]). Li and Tian [32] proved that a three-term CG algorithm has quadratic convergence with a restart strategy under some inexact line searches and the suitable assumptions.…”

On the rate of superlinear convergence of a class of variable metric methods

“…A first result concerning the rate of superlinear convergence of a particular variable metric method, the Davidon-Fletcher-Powell method [4,7], was obtained by Burmeister [3]. Assuming that the optimal step size is used he proved that this method generates a sequence which converges n-step quadratically when applied to a function F(x) depending on n variables.…”