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...

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Vydáno v:	Mathematical programming Ročník 188; číslo 2; s. 695 - 728
Hlavní autoři:	Gnegel, Fabian, Fügenschuh, Armin, Hagel, Michael, Leyffer, Sven, Stiemer, Marcus
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2021 Springer Nature B.V Springer Science + Business Media
Témata:	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
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Popis
Shrnutí:	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.
Bibliografie:	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