Continuous Piecewise Linear Delta-Approximations for Univariate Functions: Computing Minimal Breakpoint Systems
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- Rebennack, S. & Kallrath, J. J Optim Theory Appl (2015) 167: 617. doi:10.1007/s10957-014-0687-3
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For univariate functions, we compute optimal breakpoint systems subject to the condition that the piecewise linear approximator, under-, and over-estimator never deviate more than a given \(\delta \)-tolerance from the original function over a given finite interval. The linear approximators, under-, and over-estimators involve shift variables at the breakpoints allowing for the computation of an optimal piecewise linear, continuous approximator, under-, and over-estimator. We develop three non-convex optimization models: two yield the minimal number of breakpoints, and another in which, for a fixed number of breakpoints, the breakpoints are placed such that the maximal deviation is minimized. Alternatively, we use two heuristics which compute the breakpoints subsequently, solving small non-convex problems. We present computational results for 10 univariate functions. Our approach computes breakpoint systems with up to one order of magnitude less breakpoints compared to an equidistant approach.