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

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.

A Unifying Modeling Abstraction for Infinite-Dimensional Optimization

“…al. recently demonstrated how such basis expansion techniques can enhance the tractability of mixed-integer PDE problems relative to using traditional transcription methods [40].…”

“…This observation is not unique to the optimal control community and there exists much to be explored for InfiniteOpt formulations in their native forms throughout their respective communities in general. Some potential avenues of research for InfiniteOpt formulations include systematic initialization techniques that consider the formulation infinite domain, generalized pre-solving methods (e.g., feasibility checking), and enhanced transformations (e.g., basis function approaches used in [19] and [40]).…”

State elimination for mixed‐integer optimal control of partial differential equations by semigroup theory

“…This approach allows us to split the state solution into parts that are then combined with a smart use of convolution. In Reference 20 it was independently investigated however we provide a full mathematical proof of the method here. The resulting explicit control‐to‐state‐map is plugged into the objective and additional constraints and replaces the PDE, which no longer appears in the problem formulation.…”