Journal of Optimization Theory and Applications

, Volume 167, Issue 1, pp 102–117

Continuous Piecewise Linear Delta-Approximations for Bivariate and Multivariate Functions

Article

DOI: 10.1007/s10957-014-0688-2

Cite this article as:
Rebennack, S. & Kallrath, J. J Optim Theory Appl (2015) 167: 102. doi:10.1007/s10957-014-0688-2

Abstract

For functions depending on two variables, we automatically construct triangulations subject to the condition that the continuous, piecewise linear approximation, under-, or overestimation, never deviates more than a given \(\delta \)-tolerance from the original function over a given domain. This tolerance is ensured by solving subproblems over each triangle to global optimality. The continuous, piecewise linear approximators, under-, and overestimators, involve shift variables at the vertices of the triangles leading to a small number of triangles while still ensuring continuity over the entire domain. For functions depending on more than two variables, we provide appropriate transformations and substitutions, which allow the use of one- or two-dimensional \(\delta \)-approximators. We address the problem of error propagation when using these dimensionality reduction routines. We discuss and analyze the trade-off between one-dimensional (1D) and two-dimensional (2D) approaches, and we demonstrate the numerical behavior of our approach on nine bivariate functions for five different \(\delta \)-tolerances.

Keywords

Global optimization Nonlinear programming Mixed-integer nonlinear programming Nonconvex optimization  Error propagation 

Mathematics Subject Classification

90C26 

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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