Article
An optimal adaptive algorithm for the approximation of concave functions.
Mathematical Programming (Impact Factor: 2.09). 01/2006; 107:357366. DOI:10.1007/s1010700305027
Source: DBLP
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Article: Approximation of convex curves with application to the bicriterial minimum cost flow problem
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ABSTRACT: An approximation of an explicity or implicity given convex curve in the plane is given by two piecewise linear ‘outer’ and ‘inner’ curves. To compute these, three rules for choosing the supporting points are proposed and it is shown for two of them that the projective distance between inner and outer approximation decreases quadratically with the number of supporting points. This method is applied to approximate the efficient point curve of the Bicriterial Minimum Cost Flow Problem, which is a piecewise linear convex curve that may have an exponential number of breakpoints in the worst case.European Journal of Operational Research 02/1989; · 2.04 Impact Factor  [show abstract] [hide abstract]
ABSTRACT: In this work, we present an implementation of an infinite dimensionalFrankWolfe algorithm for solving static, fixed demand, symmetric or asymmetrictraffic equilibrium problems involving several classes of customers.Each customer class is characterized by its perception of two criteria, namelytime and money.Keywords: Traffic Assignment  Multicriterion  Variational Inequalities1 The multiclass modelIn the basic traffic assignment problem, an equilibrium corresponds to that...  [show abstract] [hide abstract]
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.Der SandwichAlgorithmus approximiert eine konvexe Funktion einer Variablen ber einem Intervall, indem er die Funktion und ihre Ableitung an einer Folge von Sttzstellen ausrechnet. Die Verbindung der Punkte ergibt eine stckweise lineare obere Approximation, und die Tangenten liefern eine stckweise lineare untere Approximation. Auf hnliche Art kann man einen konvexen Bereich der Ebene durch konvexe Polygone approximieren.Verschiedene Versionen des SandwichAlgorithmus unterscheiden sich durch die Regel, nach der sie die nchste Sttzstelle bestimmen. Wir zeigen fr vier natrliche Regeln (Intervallhalbierung, Steigungshalbierung, maximalerFehlerRegel und Sehnenregel), da der globale Approximationsfehler mit der Anzahln der Sttzstellen mit der bestmglichen OrdnungO(1/n 2) abnimmt.Computing 01/1992; 48(3):337361. · 0.81 Impact Factor
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