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Approximation of Non-linear Functions in Mixed Integer Programming. TU Darmstadt. Alexander Martin. Workshop on Integer Programming and Continuous Optimization Chemnitz, November 7-9, 2004. Joint work with Markus Möller and Susanne Moritz. - PowerPoint PPT Presentation
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Approximation of Non-linear Functions in Mixed Integer Programming
Alexander MartinTU Darmstadt
Workshop on Integer Programming and Continuous Optimization
Chemnitz, November 7-9, 2004
Joint work with Markus Möller and Susanne Moritz
A. Martin
2
1. Non-linear Functions in MIPs- design of sheet metal
- gas optimization
- traffic flows
2. Modelling Non-linear Functions- with binary variables
- with SOS constraints
3. Polyhedral Analysis
4. Computational Results
Outline
A. Martin
3
1. Non-linear Functions in MIPs- design of sheet metal
- gas optimization
- traffic flows
2. Modelling Non-linear Functions- with binary variables
- with SOS constraints
3. Polyhedral Analysis
4. Computational Results
Outline
A. Martin
4
Design of Transport Channels
q0
EI wmax
w(x)
l
z
x
2y z dA
40
maxy
q ll 5w w
2 384 E
4 340
y
q l x x xw(x) 2
24 E l l l
S
z
y x
dA
z
0
gq 2,60
mm
q0
EI wmax
w(x)
l
z
x
2y z dA
40
maxy
q ll 5w w
2 384 E
4 340
y
q l x x xw(x) 2
24 E l l l
S
z
y x
dA
zS
z
y x
dA
z
0
gq 2,60
mm
- Bounds on theperimeters
- Bounds on thearea(s)
- Bounds on thecentre of gravity
Goal
Subject To
Maximize stiffness
Variables - topology- material
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- contracts- physical constraints
Goal
Subject To
Minimize fuel gas consumption
Optimization of Gas Networks
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Gas Network in Detail
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Gas Networks: Nature of the Problem
• Non-linear- fuel gas consumption of compressors- pipe hydraulics- blending, contracts
• Discrete- valves- status of compressors- contracts
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8
Pressure Loss in Gas Networks
stationarycase
horizontalpipes
pout
pin
q
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9
1. Non-linear Functions in MIPs- design of sheet metal
- gas optimization
- traffic flows
2. Modelling Non-linear Functions- with binary variables
- with SOS constraints
3. Polyhedral Analysis
4. Computational Results
Outline
A. Martin
10
Approximation of Pressure Loss: Binary Approach
pin
pout
q
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Approximation of Pressure Loss: SOS Approach
pout
pin
q
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Branching on SOS Constraints
31i
1 i 1 i
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1. Non-linear Functions in MIPs- design of sheet metal
- gas optimization
- traffic flows
2. Modelling Non-linear Functions- with binary variables
- with SOS constraints
3. Polyhedral Analysis
4. Computational Results
Outline
A. Martin
14
The SOS Constraints: General Definition
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The SOS Constraints: Special Cases
• SOS Type 2 constraints
• SOS Type 3 constraints
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The Binary Polytope
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The Binary Polytope: Inequalities
21i
21iy
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The SOS Polytope
Pipe 1 Pipe 2
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|Y| Vertices FacetsMax. Coeff.
8 12 16 18 25
16 18 49 47 42
24 24 73 90 670
32 32 142 10492 50640
The SOS Polytope: Increasing Complexity
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The SOS Polytope: Properties
Theorem. There exist only polynomially many
vertices
• The vertices can be determined algorithmically• This yields a polynomial separation algorithm by solving for given and
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The SOS Polytope: Generalizations
• Pipe to pipe with respect to pressure and flow• Several pipes to several pipes• Pipes to compressors (SOS constraints of Type 4)• General Mixed Integer Programs:
Consider Ax=b and a set I of SOS constraints of Type for such that each variable is contained in exactly one SOS constraint. If the rank of A (incl. I) and are fixed then
has only polynomial many vertices.
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Binary versus SOS Approach
• Binary- more (binary) variables- more constraints- complex facets- LP solutions with fractional y variables and correct variables
• SOS+ no binary variables+ triangle condition can be incorporated within branch & bound+ underlying polyhedra are tractable
A. Martin
23
1. Non-linear Functions in MIPs- design of sheet metal
- gas optimization
- traffic flows
2. Modelling Non-linear Functions- with binary variables
- with SOS constraints
3. Polyhedral Analysis
4. Computational Results
Outline
A. Martin
24
Computational Results
Nr of Pipes Nr of
CompressorsTotal length
of pipesTime
( = 0.05)
Time
( = 0.01)
11 3 920 1.2 sec 2.0 sec
20 3 1200 1.2 sec 9.9 sec
31 15 2200 11.5 sec 104.4 sec
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