A solution framework for linear PDE-constrained mixed-integer problems

We present a general numerical solution method for control problems with state variables defined by a linear PDE over a finite set of binary or continuous control variables. We show empirically that a naive approach that applies a numerical discretization scheme to the PDEs to derive constraints for...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Mathematical programming Jg. 188; H. 2; S. 695 - 728
Hauptverfasser: Gnegel, Fabian, Fügenschuh, Armin, Hagel, Michael, Leyffer, Sven, Stiemer, Marcus
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2021
Springer Nature B.V
Springer Science + Business Media
Schlagworte:
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Computation
Constraints
Continuity (mathematics)
Discretization
Full Length Paper
Linear programming
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Mixed integer
Numerical Analysis
State variable
Theoretical
Wildfires
65N30 Mixed integer programming
Partial differential equations
Global optimal control
90C11 Finite element
90-10 Mathematical modeling or simulation for problems pertaining to operations research and mathematical programming
Finite-element methods
Finite-difference methods
Mixed-integer linear programming
Convection-diffusion equation
ISSN:0025-5610, 1436-4646
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung:We present a general numerical solution method for control problems with state variables defined by a linear PDE over a finite set of binary or continuous control variables. We show empirically that a naive approach that applies a numerical discretization scheme to the PDEs to derive constraints for a mixed-integer linear program (MILP) leads to systems that are too large to be solved with state-of-the-art solvers for MILPs, especially if we desire an accurate approximation of the state variables. Our framework comprises two techniques to mitigate the rise of computation times with increasing discretization level: First, the linear system is solved for a basis of the control space in a preprocessing step. Second, certain constraints are just imposed on demand via the IBM ILOG CPLEX feature of a lazy constraint callback. These techniques are compared with an approach where the relations obtained by the discretization of the continuous constraints are directly included in the MILP. We demonstrate our approach on two examples: modeling of the spread of wildfire and the mitigation of water contamination. In both examples the computational results demonstrate that the solution time is significantly reduced by our methods. In particular, the dependence of the computation time on the size of the spatial discretization of the PDE is significantly reduced.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
AC02-06CH11357
USDOE
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-021-01626-1