Continuous Piecewise Linear Delta-Approximations for Bivariate and Multivariate Functions
- 232 Downloads
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.
KeywordsGlobal optimization Nonlinear programming Mixed-integer nonlinear programming Nonconvex optimization Error propagation
Mathematics Subject Classification90C26
The paper was ready and sent to the Editor-in-Chief on Sept. 26th 2012 for an evaluation; then, after several exchanges and revisions, it has been submitted to the Editorial Manager of JOTA on Oct 29th, 2014.
- 8.Geißler, B.: Towards globally optimal solutions for MINLPs by discretization techniques with applications in gas network optimization. Dissertation, Universität Erlangen-Nürnberg (2011)Google Scholar
- 11.Rebennack, S., Kallrath, J.: Continuous piecewise linear delta-approximations for univariate functions: computing minimal breakpoint systems. J. Optim. Theory Appl. (2014). doi: 10.1007/s10957-014-0687-3
- 12.Kallrath, J., Rebennack, S.: Computing area-tight piecewise linear overestimators, underestimators and tubes for univariate functions. In: Butenko, S., Floudas, C., Rassias, T. (eds.) Optimization in Science and Engineering. Springer, Berlin (2014)Google Scholar