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The convergence rate of the sandwich algorithm for approximating convex functions

Die Konvergenzrate des Sandwich-Algorithmus zur Approximation konvexer Funktionen

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Abstract

The Sandwich algorithm approximates a convex function of one variable over an interval by evaluating the function and its derivative at a sequence of points. The connection of the obtained points is a piecewise linear upper approximation, and the tangents yield a piecewise linear lower approximation. Similarly, a planar convex figure can be approximated by convex polygons.

Different versions of the Sandwich algorithm use different rules for selecting the next evaluation point. We consider four natural rules (interval bisection, slope bisection, maximum error rule, and chord rule) and show that the global approximation error withn evaluation points decreases by the order ofO(1/n 2), which is optimal.

By special examples we show that the actual performance of the four rules can be very different from each other, and we report computational experiments which compare the performance of the rules for particular functions.

Zusammenfassung

Der Sandwich-Algorithmus approximiert eine konvexe Funktion einer Variablen über einem Intervall, indem er die Funktion und ihre Ableitung an einer Folge von Stützstellen ausrechnet. Die Verbindung der Punkte ergibt eine stückweise lineare obere Approximation, und die Tangenten liefern eine stückweise lineare untere Approximation. Auf ähnliche Art kann man einen konvexen Bereich der Ebene durch konvexe Polygone approximieren.

Verschiedene Versionen des Sandwich-Algorithmus unterscheiden sich durch die Regel, nach der sie die nächste Stützstelle bestimmen. Wir zeigen für vier natürliche Regeln (Intervallhalbierung, Steigungshalbierung, maximaler-Fehler-Regel und Sehnenregel), daß der globale Approximationsfehler mit der Anzahln der Stützstellen mit der bestmöglichen OrdnungO(1/n 2) abnimmt.

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Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, A-8010, Graz, Austria

    G. Rote

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  1. G. Rote
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This work was partially supported by the Fonds zur Förderung der wissenschaftlichen Forschung, Project P7486-Phy.

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Rote, G. The convergence rate of the sandwich algorithm for approximating convex functions. Computing 48, 337–361 (1992). https://doi.org/10.1007/BF02238642

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