An optimal adaptive algorithm for the approximation of concave functions.

Mathematical Programming (Impact Factor: 2.09). 01/2006; 107:357-366. DOI: 10.1007/s10107-003-0502-7
Source: DBLP

ABSTRACT Motivated by the study of parametric convex programs, we consider approximation of concave functions by piecewise affine functions.
Using dynamic programming, we derive a procedure for selecting the knots at which an oracle provides the function value and
one supergradient. The procedure is adaptive in that the choice of a knot is dependent on the choice of the previous knots.
It is also optimal in that the approximation error, in the integral sense, is minimized in the worst case.

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    ABSTRACT: In this paper, piecewise linear upper and lower bounds for univariate convex functions are derived that are only based on function value information. These upper and lower bounds can be used to approximate univariate convex functions. Furthermore, new Sandwich algo- rithms are proposed, that iteratively add new input data points in a systematic way, until a desired accuracy of the approximation is obtained. We show that our new algorithms that use only function-value evaluations converge quadratically under certain conditions on the derivatives. Under other conditions, linear convergence can be shown. Some numeri- cal examples, including a Strategic investment model, that illustrate the usefulness of the algorithm, are given.
    Informs Journal on Computing 02/2007; · 1.37 Impact Factor

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