Mixed-Integer PDE-Constrained OptimizationIMA Workshop on Uncertainty Quantification
Pelin Cay, Bart van Bloemen Waanders, Drew Kouri,Anna Thuenen and Sven Leyffer
Lehigh University, Universitat Magdeburg, Argonne National Laboratory,and Sandia National Laboratories
February 22, 2016
Outline
1 IntroductionProblem Definition and ApplicationsSource Inversion as MIP with PDE ConstraintsProblem Classification and Challenges
2 Early Theoretical & Numerical ResultsEliminating the PDE & State VariablesNumerical Experience with Source InversionControl Regularization: Not All Norms Are EqualHeat Equation: Actuator Design
3 Rounding-Based Heuristic for Cloaking
4 Conclusions
2 / 42
Mixed-Integer PDE-Constrained Optimization (MIPDECO)PDE-constrained MIP ... u = u(t, x , y , z) ⇒ infinite-dimensional!
t is time index; x , y , z are spatial dimensionsminimize
u,wF(u,w)
subject to C(u,w) = 0u ∈ U , and w ∈ Zp (integers),
u(t, x , y , z): PDE states, controls, & design parameters
w discrete or integral variables
MIPDECO Warning
w = w(t, x , y , z) ∈ Z may beinfinite-dimensional integers!
It’s a MIP, Jim,but not as we know it!
3 / 42
Mixed-Integer PDE-Constrained Optimization (MIPDECO)PDE-constrained MIP ... u = u(t, x , y , z) ⇒ infinite-dimensional!
t is time index; x , y , z are spatial dimensionsminimize
u,wF(u,w)
subject to C(u,w) = 0u ∈ U , and w ∈ Zp (integers),
u(t, x , y , z): PDE states, controls, & design parameters
w discrete or integral variables
MIPDECO Warning
w = w(t, x , y , z) ∈ Z may beinfinite-dimensional integers!
It’s a MIP, Jim,but not as we know it!
3 / 42
Grand-Challenge Applications of MIPDECO
Topology optimization [Sigmund and Maute, 2013]
Nuclear plant design: select core types &control flow rates [Committee, 2010]
Well-selection for remediation ofcontaminated sites [Ozdogan, 2004]
Design of next-generation solar cells[Reinke et al., 2011]
Design of wind-farms [Zhang et al., 2013]
Scheduling for disaster recovery:oil-spills [You and Leyffer, 2010]
& wildfires [Donovan and Rideout, 2003]
Design & control of gas networks,[De Wolf and Smeers, 2000, Martin et al., 2006, Zavala, 2014]
... also as optimization under uncertainty
4 / 42
Uncertainty Quantification and MIPDECO
Design of experiments, e.g. discrete sensor placement
Akaike’s Information Criterion: parameter & structure est.
AIC: maximize log-likelihood & minimize nonzeros u
minimizew∈0,1
N∑k=1
ek(u)TR−1ek(u)+l∑
i=1
wi s.t. −Mwi ≤ ui ≤ Mwi
where R is known co-variance
5 / 42
Source Inversion as MIP with PDE Constraints
Simple Example: Locate number of sources to match observation u
minimizeu,w
J =1
2
∫Ω
(u − u)2dΩ least-squares fit
subject to −∆u =∑k,l
wkl fkl in Ω Poisson equation∑k,l
wkl ≤ S and wkl ∈ 0, 1 source budget
with Dirichlet boundary conditions u = 0 on ∂Ω.
E.g. Gaussian source term, σ > 0, centered at (xk , yl)
fkl(x , y) := exp
(−‖(xk , yl)− (x , y)‖2
σ2
),
Motivated by porous-media flow application to determine numberof boreholes, [Ozdogan, 2004, Fipki and Celi, 2008]
6 / 42
Source Inversion as MIP with PDE Constraints
Consider 2D example with Ω = [0, 1]2 and discretize PDE:
5-point finite-difference stencil; uniform mesh h = 1/N
Denote ui ,j ≈ u(ih, jh) approximation at grid points
minimizeu,w
Jh =h2
2
N∑i ,j=0
(ui ,j − ui ,j)2
subject to4ui ,j − ui ,j−1 − ui ,j+1 − ui−1,j − ui+1,j
h2=
N∑k,l=1
wkl fkl(ih, jh)
u0,j = uN,j = ui ,0 = ui ,N = 0N∑
k,l=1
wkl ≤ S and wkl ∈ 0, 1
⇒ finite-dimensional (convex) MIQP
7 / 42
Source Inversion as MIP with PDE Constraints
Potential source locations (blue dots) on 16× 16 meshCreate target u using red square sources
8 / 42
Source Inversion as MIP with PDE Constraints
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Contours of TARGET ubar(x,y)
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Relative error in States ubar(x,y) - u(x,y) & Source Location
-0.1
-0.05
0
0.05
0.1
0.15
Target (3 sources), reconstructed sources, & error on 32× 32 mesh
9 / 42
Outline
1 IntroductionProblem Definition and ApplicationsSource Inversion as MIP with PDE ConstraintsProblem Classification and Challenges
2 Early Theoretical & Numerical ResultsEliminating the PDE & State VariablesNumerical Experience with Source InversionControl Regularization: Not All Norms Are EqualHeat Equation: Actuator Design
3 Rounding-Based Heuristic for Cloaking
4 Conclusions
10 / 42
Mixed-Integer PDE-Constrained Optimization (MIPDECO)
minimize
u,wF(u,w)
subject to C(u,w) = 0u ∈ U , and w ∈ Zp (integers),
u(t, x , y , z): PDE states, controls, & design parameters
w discrete or integral variables
Towards a problem characterization
Type of PDE: different classes of PDEse.g. elliptic, parabolic, hyperbolic, nonlinear, ...
Class of Integers: binary, general integers, etc
Type of Objective: functional form of objective
Type of Constraints: characterize c/s other than PDE
Discretization: discretization method & CUTEr classification
11 / 42
Mesh-Independent & Mesh-Dependent Integers
Definition (Mesh-Independent & Mesh-Dependent Integers)
1 The integer variables are mesh-independent, iff number ofinteger variables is independent of the mesh.
2 The integer variables are mesh-dependent, iff the number ofinteger variables depends on the mesh.
Mesh-Independent
Manageable treeTheory possible
Mesh-Dependent
Exploding tree sizeTheory???
12 / 42
Theoretical Challenges of MIPDECO
Functional Analysis (mesh-dependent integers)
Denis Ridzal: What function space is w(x , y) ∈ 0, 1?
Consistently approximate w(x , y) ∈ 0, 1 as h→ 0?
Conjecture: w(x , y) ∈ 0, 1 6= L2(Ω)... e.g. binary support of Cantor set not integrable
Likely need additional regularity assumptions
Coupling between Discretization & Integers
Discretization scheme (e.g. upwinding for wave equation) dependson direction of flow (integers).
Application: gas network models with flow reversals. . . open postdoc position at Argonne!
13 / 42
Computational Challenges of MIPDECO
Approaches for humongous branch-and-bound trees... e.g. 3D topology optimization with 109 binary variables
Warm-starts for PDE-constrained optimization (nodes)
Guarantees for nonconvex (nonlinear) PDE constraints... factorable programming approach hopeless for 109 vars!
log
^
31
2
2x x
x
*
+
... f (x1, x2) = x1 log(x2) + x32
14 / 42
MIPDECO: Two Cultures Collide
Observation
PDE-optimization & MIP developed separately⇒ different assumptions, methodologies, and
computational kernels!
PDE-Optimization Mixed-Integer Programming
Obtain good solutions efficiently Deliver certificate of optimality
Nonlinear optimization:Newton’s method
Combinatorial optimization:branch-and-cut
Iterative Krylov solvers Factors & rank-one updates
Run on bleeding-edge HPC Limited HPC developments
Potential for Disaster, or Opportunity for Innovation!
15 / 42
Outline
1 IntroductionProblem Definition and ApplicationsSource Inversion as MIP with PDE ConstraintsProblem Classification and Challenges
2 Early Theoretical & Numerical ResultsEliminating the PDE & State VariablesNumerical Experience with Source InversionControl Regularization: Not All Norms Are EqualHeat Equation: Actuator Design
3 Rounding-Based Heuristic for Cloaking
4 Conclusions
16 / 42
Source Inversion as MIP with PDE Constraints
Find number and location of sources to match observation u
minimizeu,w
J =1
2
∫Ω
(u(w)− u)2dΩ least-squares fit
subject to −∆u =∑k,l
wkl fkl in Ω Poisson equation∑k,l
wkl ≤ S and wkl ∈ 0, 1 source budget
MIP with convex quadratic objective on Ω = [0, 1]2
5-point finite-difference stencil; uniform mesh h = 1/N
Denote ui ,j ≈ u(ih, jh) approximation at grid points
17 / 42
Cool MIPDECO Trick: Eliminating the PDE
Discretized PDE constraint (Poisson equation)
4ui ,j − ui ,j−1 − ui ,j+1 − ui−1,j − ui+1,j
h2=∑k,l
wkl fkl(ih, jh), ∀i , j
⇔ Au =∑
wkl fkl , where wkl ∈ 0, 1 only appear on RHS!
Elimination of PDE and states u(x , y , z)
Au =∑k,l
wkl fkl ⇔ u = A−1
∑k,l
wkl fkl
=∑k,l
wklA−1fkl
Solve n2 2n PDEs: u(kl) := A−1fkl
Eliminate u =∑
k,l wklu(kl) from Source Inversion
18 / 42
Cool MIPDECO Trick: Eliminating the PDE
Eliminating u =∑
k,l wklu(kl) in MINLP gives:
minimizew
Jh =h2
2
N∑i ,j=0
∑k,l
wklu(kl)ij − ui ,j
2
subject toN∑
k,l=1
wkl ≤ S and wkl ∈ 0, 1
Eliminates the states u (N2 variables)
Eliminates the PDE constraint (N2 constraints)
... generalizes to other PDEs (with integer controls on RHS)
Simplified model is quadratic knapsack problem
19 / 42
Outline
1 IntroductionProblem Definition and ApplicationsSource Inversion as MIP with PDE ConstraintsProblem Classification and Challenges
2 Early Theoretical & Numerical ResultsEliminating the PDE & State VariablesNumerical Experience with Source InversionControl Regularization: Not All Norms Are EqualHeat Equation: Actuator Design
3 Rounding-Based Heuristic for Cloaking
4 Conclusions
20 / 42
Numerical Experience with Source Inversion
Find number and location of sources to match observation u
minimizeu,w
J =1
2
∫Ω
(u(w)− u)2dΩ least-squares fit
subject to −∆u =∑k,l
wkl fkl in Ω Poisson equation∑k,l
wkl ≤ S and wkl ∈ 0, 1 source budget
MIP with convex quadratic objective
Computational Experiments:
1 Test NLP-plus-rounding heuristic versus MINLP2 Effect of mesh-dependent vs. mesh-independent integers
Mesh-independent: pick sources from 36 potential locationsMesh-dependent: all nodes are potential locations
3 Effect of state-elimination trick
21 / 42
1st Example Mixed-Integer PDE-Constrained Optimization
Potential source locations (blue dots) on 16× 16 meshCreate target u using red square sources
22 / 42
Approach 1: NLP-Solve, Knapsack Rounding, and MIP
Knapsack Rounding
1 Solve continuous relaxation using NLP solver
2 Round largest S locations, wi , to one & set all others to zero
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Error in States ubar(x,y) - u(x,y) & Source Location ×10
-3
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Error in States ubar(x,y) - u(x,y) & Source Location ×10
-3
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Knapsack-rounded NLP (left) and MINLP (right)
MINLP solution better: NLP-err = 0.0388 > 0.0307 = MIP-err
23 / 42
Approach 1: NLP-Solve, Knapsack Rounding, and MIP
Knapsack Rounding
1 Solve continuous relaxation using NLP solver
2 Round largest S locations, wi , to one & set all others to zero
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Error in States ubar(x,y) - u(x,y) & Source Location ×10
-3
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Error in States ubar(x,y) - u(x,y) & Source Location ×10
-3
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Knapsack-rounded NLP (left) and MINLP (right)
MINLP solution better: NLP-err = 0.0388 > 0.0307 = MIP-err23 / 42
Mesh-Independent Source Inversion: MINLP Solvers
Number of Nodes and CPU time for Increasing Mesh Sizes
Mesh-Size
0 10 20 30 40 50 60 70
No
de
s
102
103
104
105
BonminOA
MINLP
Minotaur
Mesh-Size
0 10 20 30 40 50 60 70
CP
U T
ime
100
101
102
103
104
105
BonminOA
MINLP
Minotaur
Number of Nodes independent of mesh size!
MINLP & Minotaur: filterSQP runs out of memory for N ≥ 32
BonminOA takes roughly 100 iterations ... quadratic objective
24 / 42
Mesh-Dependent (all) Source Inversion: MINLP Solvers
Number of Nodes and CPU time for Increasing Mesh Sizes
Mesh-Size
4 6 8 10 12 14 16
No
de
s
101
102
103
104
105
106
107
BonminOA
BonminBB
MINLP
Minotaur
Mesh-Size
4 6 8 10 12 14 16
CP
U T
ime
10-2
100
102
104
106
108
BonminOA
BonminBB
MINLP
Minotaur
Number of nodes explodes with mesh size!
OA <BREAK> after 130,000 seconds
25 / 42
Elimination of States & PDEs: Source Inversion
CPU Time for Increasing Mesh Sizes: Simplified vs. Original Model
Mesh-Size
5 10 15 20 25 30 35
CP
U T
ime
10 -2
10 -1
10 0
10 1
10 2
10 3
10 4
MINLP
MINLP-Simple
Eliminating PDEs is two orders of magnitude faster!
26 / 42
Elimination of States & PDEs: Source Inversion
CPU Time for Increasing Mesh Sizes: Simplified vs. Original Model
8× 8 16× 16 32× 32
Presolve Time 0.05 1.30 62.51Simplified Model 0.18 0.50 2.38
Total Simplified 0.23 1.80 64.89
Full PDE Model 2.10 29.43 1013.21
... using NLP solve for PDE (inefficient)
Presolve is cheap ... simplified model solves much faster!
27 / 42
First Conclusions: Source Inversion
Numerical Results
Solve mesh-independent problems with coarse discretization
Mesh-dependent instances cannot be solved
Outer Approximation (Bon-OA) inefficient for these instances
Trick # 1: elimination of states and PDE constraint
Nonlinear solvers run into storage issues
... not surprising: MIPDECO trees grow like tribbles!
28 / 42
First Conclusions: Source Inversion
Numerical Results
Solve mesh-independent problems with coarse discretization
Mesh-dependent instances cannot be solved
Outer Approximation (Bon-OA) inefficient for these instances
Trick # 1: elimination of states and PDE constraint
Nonlinear solvers run into storage issues
... not surprising: MIPDECO trees grow like tribbles!
28 / 42
Outline
1 IntroductionProblem Definition and ApplicationsSource Inversion as MIP with PDE ConstraintsProblem Classification and Challenges
2 Early Theoretical & Numerical ResultsEliminating the PDE & State VariablesNumerical Experience with Source InversionControl Regularization: Not All Norms Are EqualHeat Equation: Actuator Design
3 Rounding-Based Heuristic for Cloaking
4 Conclusions
29 / 42
Control Regularization: Not All Norms Are Equal
Poisson with Distributed Control [OPTPDE, 2014] & [Troltzsch, 1984]minimize
u,w‖u − ud‖2
L2(Ω) +
∫ΓeΓ u ds + α ‖w‖2
Lx
subject to −∆u + u = w + eΩ in Ω∂u
∂n= 0 on boundary Γ
w(t) ∈ 0, 1
L1 or L2 regularization term for control w(t) ∈ 0, 1?
Good Norms for MIPs
MIP’ers prefer polyhedral norms ... promote integrality
Old MIP trick: w2(t) = |w(t)| for w(t) ∈ 0, 1⇒ L1-norm same as L2-norm on binary variables!
30 / 42
Not All Norms Are Equal
Consider Distributed Control for increasing mesh-size
CPU for L2 Regularization
Mesh Minotaur B-BB B-Hyb B-OA
8x8 0.04 0.80 2.54 126.8116x16 6.61 72.21 1305.00 Time32x32 Time Time Time Time
CPU for L1 Regularization
Mesh Minotaur B-BB B-Hyb B-OA
8x8 0.03 0.48 0.21 0.0416x16 0.11 3.62 0.66 0.2032x32 0.18 62.66 3.53 0.74
L1 regularization is equivalent to L2, but faster
Many fewer nodes in tree-searches ⇒ solve up to 256× 256
31 / 42
Not All Norms Are Equal
Consider Distributed Control for increasing mesh-size
CPU for L2 Regularization
Mesh Minotaur B-BB B-Hyb B-OA
8x8 0.04 0.80 2.54 126.8116x16 6.61 72.21 1305.00 Time32x32 Time Time Time Time
CPU for L1 Regularization
Mesh Minotaur B-BB B-Hyb B-OA
8x8 0.03 0.48 0.21 0.0416x16 0.11 3.62 0.66 0.2032x32 0.18 62.66 3.53 0.74
L1 regularization is equivalent to L2, but faster
Many fewer nodes in tree-searches ⇒ solve up to 256× 256
31 / 42
Outline
1 IntroductionProblem Definition and ApplicationsSource Inversion as MIP with PDE ConstraintsProblem Classification and Challenges
2 Early Theoretical & Numerical ResultsEliminating the PDE & State VariablesNumerical Experience with Source InversionControl Regularization: Not All Norms Are EqualHeat Equation: Actuator Design
3 Rounding-Based Heuristic for Cloaking
4 Conclusions
32 / 42
Problem 2: Actuator Placement and Operation [Falk Hante]
Goal: Control temperature with actuators
Select sequence of control inputs (actuators)
Choose continuous control (heat/cool) at locations
Match prescribed temperature profile
... “de-mist bathroom mirror with hair-drier”
Potential Actuator Locations l = 1, . . . , L
33 / 42
Problem 2: Actuator Placement and OperationFind optimal sequence of actuators, wl(t), and controls, vl(t):
minimizeu,v ,w
‖u(tf , ·)‖2Ω + 2‖u‖2
T×Ω + 1500‖v‖
2T
subject to∂u
∂t− κ∆u =
L∑l=1
vl(t)fl in T × Ω
wl(t) ∈ 0, 1,L∑
l=1
wl(t) ≤W , ∀t ∈ T
Lwl(t) ≤ vl(t) ≤ Uwl(t), ∀l = 1, . . . , L, ∀t ∈ T
where
fl(x , y) =1√2πσ
exp
(−‖(x , y)− (xl , yl)‖2)
2σ
)point-source for actuators at (xl , yl) ... movies!
34 / 42
Outline
1 IntroductionProblem Definition and ApplicationsSource Inversion as MIP with PDE ConstraintsProblem Classification and Challenges
2 Early Theoretical & Numerical ResultsEliminating the PDE & State VariablesNumerical Experience with Source InversionControl Regularization: Not All Norms Are EqualHeat Equation: Actuator Design
3 Rounding-Based Heuristic for Cloaking
4 Conclusions
35 / 42
Topology Design of Cloaking Devices/Scatterers
Design of cloaking device on domain Ω
Cloak subdomain Ω0 (red dashes) bypreventing (complex) wave fromentering domain
Design scatterer in subdomain Ω...w(x , y) ∈ 0, 1PDE: 2D Helmholtz (over C) withRobin boundary conditions
Incident wave is exp(ik0y) forwavelength k0 = 6π
where i =√−1
Romulan Warbird
Scatterer
36 / 42
Topology Design of Cloaking Devices/Scatterers
Control: w = w(x , y) in ΩStates: u = u(x , y) in ΩTarget: u0 = u0(x , y) in Ω0
minimizeu,v ,w
J(u) = 12‖u + u0‖2
2,Ω0
subject to −∆u − k20 (1 + qw)u = k2
0qwu0 in Ω∂u∂n − ik0u = 0 on ∂Ω
w ∈ 0, 1 in Ω.
Discretization: finite-differences with l = 3 nodes per scatterelement, w(x , y).
37 / 42
Strip Rounding Heuristic
Cannot solve on reasonable mesh/domain with any MINLP solver.
Algorithm: Strip Rounding HeuristicSolve continuous relaxation & initialize i = 1for i=1,...,N do
Round a strip w(xi , yj) for all jResolve relaxation with w(xk , y) fixed for all k ≤ i
end
Round fractional w(x , y) following direction of wave
38 / 42
Strip Rounding Heuristic
Cannot solve on reasonable mesh/domain with any MINLP solver.
Algorithm: Strip Rounding HeuristicSolve continuous relaxation & initialize i = 1for i=1,...,N do
Round a strip w(xi , yj) for all jResolve relaxation with w(xk , y) fixed for all k ≤ i
end
Round fractional w(x , y) following direction of wave
38 / 42
Strip Rounding Heuristic
Cannot solve on reasonable mesh/domain with any MINLP solver.
Algorithm: Strip Rounding HeuristicSolve continuous relaxation & initialize i = 1for i=1,...,N do
Round a strip w(xi , yj) for all jResolve relaxation with w(xk , y) fixed for all k ≤ i
end
Round fractional w(x , y) following direction of wave
38 / 42
Results for Strip Rounding
Scatterer, w(x , y) States u(x , y)
... resolve PDE on finer mesh for fixed controls
39 / 42
... Solution Not Physical!
Coarse States Resolved States
... not clear we’re getting the correct physics!
40 / 42
Conclusions
Mixed-Integer PDE-Constrained Optimization (MIPDECO)
Class of challenging problems with important applications
Subsurface flow: oil recovery or environmental remediationDesign and operation of gas-/power-networks
Classification: mesh-dependent vs. mesh-independent
On-going work: Building library of test problems... formulation matters: interplay of binary and continuous
Elimination of PDE and state variables u(t, x , y , z)
Discretized PDEs ⇒ huge MINLPs ... push solvers to limit
Outlook and Extensions
Consider multi-level in space (network) and time
Move toward truly multi-level approach similar to PDEs
Interested in new UQ applications involving MIP & PDEs ...
41 / 42
Our five-year mission
To boldly go where no optimizer has gone before ...
... to explore strange new PDEs & MIPs!
42 / 42
Committee, T. (2010).Advanced fueld pellet materials and fuel rod design for water cooled reactors.Technical report, International Atomic Energy Agency.
De Wolf, D. and Smeers, Y. (2000).The gas transmission problem solved by an extension of the simplex algorithm.Management Science, 46:1454–1465.
Donovan, G. and Rideout, D. (2003).An integer programming model to optimize resource allocation for wildfirecontrainment.Forest Science, 61(2).
Fipki, S. and Celi, A. (2008).The use of multilateral well designs for improved recovery in heavy oil reservoirs.In IADV/SPE Conference and Exhibition, Orlanda, Florida. SPE.
Martin, A., Moller, M., and Moritz, S. (2006).Mixed integer models for the stationary case of gas network optimization.Mathematical Programming, 105:563–582.
OPTPDE (2014).OPTPDE — a collection of problems in PDE-constrained optimization.http://www.optpde.net.
Ozdogan, U. (2004).Optimization of well placement under time-dependent uncertainty.Master’s thesis, Stanford University.
42 / 42
Reinke, C. M., la Mata Luque, T. M. D., Su, M. F., Sinclair, M. B., andEl-Kady, I. (2011).Group-theory approach to tailored electromagnetic properties of metamaterials:An inverse-problem solution.Physical Review E, 83(6):066603–1–18.
Sigmund, O. and Maute, K. (2013).Topology optimization approaches: A comparative review.Structural and Multidisciplinary Optimization, 48(6):1031–1055.
Troltzsch, F. (1984).The generalized bang-bang-principle and the numerical solution of a parabolicboundary-control problem with constraints on the control and the state.Zeitschrift fur Angewandte Mathematik und Mechanik, 64(12):551–556.
You, F. and Leyffer, S. (2010).Oil spill response planning with MINLP.SIAG/OPT Views-and-News, 21(2):1–8.
Zavala, V. M. (2014).Stochastic optimal control model for natural gas networks.Computers & Chemical Engineering, 64:103–113.
Zhang, P., Romero, D., Beck, J., and Amon, C. (2013).Integration of AI and OR Techniques in Contraint Programming forCombinatorial Optimization Problems, chapter Solving Wind Farm LayoutOptimization with Mixed Integer Programming and Constraint Programming.Springer Verlag.
42 / 42
