MIPDECO
Mixed-Integer PDE Constrained Optimization (MIPDECO) Test Problems
This page links to sets of MIPDECO test problems written in AMPL and described in a companion report.
Each problem includes a characterization in terms of type of PDE, class of integers, type of objective, type of constraints, and discretization scheme.
A tar-ball with all ampl models, data, and command files is available here. A tar-ball with the corresponding stub.nl files is available here.
Laplace Source Inversion
Problem is to identify source to match an observed state, governed by Laplace equation with Dirichtlet boundary conditions.
| Type of PDE | Laplace equation on [0,1]^2 with Dirichlet boundary conditions |
| Class of Integers | Mesh-dependent & mesh-independent binary variables |
| Type of Objective | Least-squares (inverse problem) with possible regularization term |
| Type of Constraints | Knapsack constraint on binary variables |
| Discretization | Five-point finite difference stencil |
| NAME | mod file | dat file | mesh | optimal objective |
|---|---|---|---|---|
| SrcDetct-MeshDep-008-1 | SrcDetct.mod | MeshDep-008-1.dat | 8x8 | ??? |
| SrcDetct-MeshDep-008-2 | SrcDetct.mod | MeshDep-008-2.dat | 8x8 | ??? |
| SrcDetct-MeshDep-008-3 | SrcDetct.mod | MeshDep-008-3.dat | 8x8 | ??? |
| SrcDetct-MeshDep-016-1 | SrcDetct.mod | MeshDep-016-1.dat | 16x16 | ??? |
| SrcDetct-MeshDep-016-2 | SrcDetct.mod | MeshDep-016-2.dat | 16x16 | ??? |
| SrcDetct-MeshDep-016-3 | SrcDetct.mod | MeshDep-016-3.dat | 16x16 | ??? |
| SrcDetct-MeshDep-032-1 | SrcDetct.mod | MeshDep-032-1.dat | 32x32 | ??? |
| SrcDetct-MeshDep-032-2 | SrcDetct.mod | MeshDep-032-2.dat | 32x32 | ??? |
| SrcDetct-MeshDep-032-3 | SrcDetct.mod | MeshDep-032-3.dat | 32x32 | ??? |
| SrcDetct-MeshInd-008-1 | SrcDetct.mod | MeshInd-008-1.dat | 8x8 | ??? |
| SrcDetct-MeshInd-008-2 | SrcDetct.mod | MeshInd-008-2.dat | 8x8 | ??? |
| SrcDetct-MeshInd-008-3 | SrcDetct.mod | MeshInd-008-3.dat | 8x8 | ??? |
| SrcDetct-MeshInd-016-1 | SrcDetct.mod | MeshInd-016-1.dat | 16x16 | ??? |
| SrcDetct-MeshInd-016-2 | SrcDetct.mod | MeshInd-016-2.dat | 16x16 | ??? |
| SrcDetct-MeshInd-016-3 | SrcDetct.mod | MeshInd-016-3.dat | 16x16 | ??? |
| SrcDetct-MeshInd-032-1 | SrcDetct.mod | MeshInd-032-1.dat | 32x32 | ??? |
| SrcDetct-MeshInd-032-2 | SrcDetct.mod | MeshInd-032-2.dat | 32x32 | ??? |
| SrcDetct-MeshInd-032-3 | SrcDetct.mod | MeshInd-032-3.dat | 32x32 | ??? |
In this problem, the state variables, u, can be eliminated, resulting in a pure integer QP. The model that defines the problems without state variables is SrcDetctElim.mod, and the AMPL script, Presolve.ampl implements the presolve step. The calling sequence is
ampl: model SrcDetctElim.mod; data MeshDep-008-1.dat; include Presolve.ampl;
for the small example, SrcDetct-MeshDep-008-1.
Distributed Control with Neumann Boundary Conditions
Parabolic Robin Boundary Problem in One Spatial Dimension
This model is modified from OPTPDE library (http://www.optpde.uni-hamburg.de/result.php?id=8). The goal is to find the optimal state and control variables that satisfy the Heat equation with boundary conditions. There are two classes of models: one with binary controls, and one with controls in {-1,0,1}. Both L1 and L2 regularizations are used.
| Type of PDE | Heat equation on [0,1]x[0,1.58] with Robin and Neuman boundary conditions |
| Class of Integers | Mesh-dependent & mesh-independent binary variables |
| Type of Objective | Least-squares (inverse problem) with L1 or L2 regularization term |
| Type of Constraints | Binary or integer controls |
| Discretization | Crank-Nicolson formula |
| NAME | mod file | dat file | mesh | optimal objective |
|---|---|---|---|---|
| RobinBin-08 | RobinBin.mod | m08.dat | 8x8 | 0.0305 |
| RobinBin-16 | RobinBin.mod | m16.dat | 16x16 | 0.0190 |
| RobinBin-32 | RobinBin.mod | m32.dat | 32x32 | 0.0145 |
| RobinBinL1-08 | RobinBinL1.mod | m08.dat | 8x8 | 0.0305 |
| RobinBinL1-16 | RobinBinL1.mod | m16.dat | 16x16 | 0.0190 |
| RobinBinL1-32 | RobinBinL1.mod | m32.dat | 32x32 | 0.0145 |
| RobinInt-08 | RobinInt.mod | m08.dat | 8x8 | 0.0305 |
| RobinInt-16 | RobinInt.mod | m16.dat | 16x16 | 0.0190 |
| RobinInt-32 | RobinInt.mod | m32.dat | 32x32 | 0.0145 |
| RobinIntL1-08 | RobinIntL1.mod | m08.dat | 8x8 | 0.0305 |
| RobinIntL1-16 | RobinIntL1.mod | m16.dat | 16x16 | 0.0190 |
| RobinIntL1-32 | RobinIntL1.mod | m32.dat | 32x32 | 0.0145 |