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Journal of Optimization Theory and Applications

, Volume 167, Issue 2, pp 617–643 | Cite as

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

  • Steffen Rebennack
  • Josef Kallrath
Article

Abstract

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.

Keywords

Global optimization Nonlinear programming Mixed-integer nonlinear programming Non-convex optimization 

Mathematics Subject Classification

90C26 

Notes

Acknowledgments

We thank Jan Jagla (GAMS GmbH, Cologne) for discussions on bi-level programs and Dr. Alexander Mitsos (MIT, Boston) for his favorable comments related to the SIP nature of our problem, Timo Lohmann and Greg Steeger (both Colorado School of Mines) for their careful proof-reading.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Division of Economics and BusinessColorado School of MinesGoldenUSA
  2. 2.Department of AstronomyUniversity of FloridaGainesvilleUSA
  3. 3.BASF SEScientific ComputingLudwigshafenGermany

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