Computing Area-Tight Piecewise Linear Overestimators, Underestimators and Tubes for Univariate Functions
We develop algorithms computing piecewise linear, continuous overestimators, underestimators and tubes for univariate functions which minimize the area between the approximator and the function. We call such approximators area-tight. Area-tight overestimators, underestimators and tubes are of interest when solving large-scale mixed-integer non-linear programming problems (MINLPs). The function to be approximated is typically non-linear and non-convex; it should be univariate and discontinuous only at a finite number of points. The number of breakpoints for the approximation as well as the absolute allowable deviation of the approximator and the original function are input for the algorithms. The presented algorithms are fully automatic and make no assumptions on the shape of the function to be approximated. We provide computational tests for ten functions.
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