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

Fabian Gnegel,Armin Fügenschuh,Sven Leyffer,Michael Hagel,Marcus Stiemer

doi:10.1007/s10107-021-01626-1

Abstract

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.

Highlights

When a model in engineering or other applied sciences yields multiple feasible parameter choices, one naturally asks, what the optimal choice is for the given application in mind
After the formulation of the general model (Sect. 2), we provide some basic concepts both of the discrete optimization and of the numerics of partial differential equations that are required for deriving our solution scheme
While mixed-integer linear program (MILP) solvers have achieved a great degree of success, their intrinsic properties make them less suitable for mixed-integer PDE-constrained optimization” (MIPDECO), as we demonstrate below

Summary

Introduction

When a model in engineering or other applied sciences yields multiple feasible parameter choices, one naturally asks, what the optimal choice is for the given application in mind. Often these problems are formulated as optimization problems, where the parameters are used as variables and optimality is specified as the minimum (maximum) of an objective function. Problems of very different nature, such as the optimal control of a chemical plant or the search of the shortest path to the closest train station on a map, can be formulated as optimization problems The former requires the control of a complicated engineering problem, sometimes modeled by some form of differential equation, while the latter is purely combinatorial. A more indepth study of the methods and concepts can be found in the references given in the particular sections

Objectives

Methods

Conclusion