Continuous Piecewise Linear Delta-Approximations for Univariate Functions: Computing Minimal Breakpoint Systems

• Published in: Journal of optimization theory and applications Vol. 167; no. 2; pp. 617 - 643
• Main Authors: Rebennack, Steffen, Kallrath, Josef
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.11.2015; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Approximation; Calculus of Variations and Optimal Control; Optimization; Computation; Convex analysis; Deviation; Engineering; Heuristic; Intervals; Linear programming; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Nonlinear programming; Operations Research/Decision Theory; Optimization; Production planning; Studies; Texts; Theory of Computation; Global optimization; Non-convex optimization; 90C26; Nonlinear programming; Mixed-integer nonlinear programming
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-014-0687-3

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 δ -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.