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

MIPDECOlib: Parabolic Robin Boundary Problem in One Spatial Dimension

NAME	mod file	dat file	mesh	optimal objective
SrcDetct-MeshDep-008-1	SrcDetct.mod	MeshDep-008-1.dat	8x8	0.0022
SrcDetct-MeshDep-008-2	SrcDetct.mod	MeshDep-008-2.dat	8x8	0.0001
SrcDetct-MeshDep-008-3	SrcDetct.mod	MeshDep-008-3.dat	8x8	0.0001
SrcDetct-MeshDep-016-1	SrcDetct.mod	MeshDep-016-1.dat	16x16	0.0076
SrcDetct-MeshDep-016-2	SrcDetct.mod	MeshDep-016-2.dat	16x16	0.0000
SrcDetct-MeshDep-016-3	SrcDetct.mod	MeshDep-016-3.dat	16x16	0.5675
SrcDetct-MeshDep-032-1	SrcDetct.mod	MeshDep-032-1.dat	32x32	0.0298
SrcDetct-MeshDep-032-2	SrcDetct.mod	MeshDep-032-2.dat	32x32	0.0000
SrcDetct-MeshDep-032-3	SrcDetct.mod	MeshDep-032-3.dat	32x32	2.2180
SrcDetct-MeshInd-008-1	SrcDetct.mod	MeshInd-008-1.dat	8x8	0.0002
SrcDetct-MeshInd-008-2	SrcDetct.mod	MeshInd-008-2.dat	8x8	0.0001
SrcDetct-MeshInd-008-3	SrcDetct.mod	MeshInd-008-3.dat	8x8	0.0001
SrcDetct-MeshInd-016-1	SrcDetct.mod	MeshInd-016-1.dat	16x16	0.0006
SrcDetct-MeshInd-016-2	SrcDetct.mod	MeshInd-016-2.dat	16x16	0.0004
SrcDetct-MeshInd-016-3	SrcDetct.mod	MeshInd-016-3.dat	16x16	0.3271
SrcDetct-MeshInd-032-1	SrcDetct.mod	MeshInd-032-1.dat	32x32	0.0023
SrcDetct-MeshInd-032-2	SrcDetct.mod	MeshInd-032-2.dat	32x32	0.0015
SrcDetct-MeshInd-032-3	SrcDetct.mod	MeshInd-032-3.dat	32x32	1.3068

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.

The Mother Problem: Distributed Control with Neumann Boundary Conditions

The objective is to find optimal control and state variables for an optimal control problem that has Poisson equation with a potential term and Neumann boundary conditions. Control variables are treated as binary variables.

Source: Continuous form taken from OPTPDE library Problem ccdist1 submitted by Roland Herzog, see http://www.optpde.uni-hamburg.de/result.php?id=2

Type of PDE	Poisson equation with a potential term on [01]x[0,1] with 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	Five-point finite-difference stencil

MIPDECOlib: Parabolic Robin Boundary Problem in One Spatial Dimension

NAME	mod file	dat file	mesh	optimal objective
MotherProbL1-md008	MotherProbL1.mod	md008.dat	8x8	1274.4697
MotherProbL1-md016	MotherProbL1.mod	md016.dat	16x16	1148.5152
MotherProbL1-md032	MotherProbL1.mod	md032.dat	32x32	1087.0657
MotherProbL2-md008	MotherProbL2.mod	md008.dat	8x8	1274.4697
MotherProbL2-md016	MotherProbL2.mod	md016.dat	16x16	1148.5152
MotherProbL2-md032	MotherProbL2.mod	md032.dat	32x32	1087.0657 (?)

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

MIPDECOlib: Parabolic Robin Boundary Problem in One Spatial Dimension

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