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

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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

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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

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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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Pressure Loss in Gas Networks

stationarycase

horizontalpipes

pout

pin

q

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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

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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

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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

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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

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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