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Martin Grötschel Institut für Mathematik, Technische Universität Berlin (TUB) DFG-Forschungszentrum „Mathematik für Schlüsseltechnologien“ (MATHEON) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)
[email protected] http://www.zib.de/groetschel
From Simulation to Optimization
Martin Grötschel Workshop
MANAGEMENT OF GAS NETWORKS mathematical solution technologies
Dutch Embassy, Berlin, October 15, 2011
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Who am I? Currently:
Mathematics professor at TU Berlin
Vice president of the Zuse Institute Berlin (ZIB)
Formerly:
Chair of the DFG Research Center MATHEON
President of the German Mathematical Society
More important in this context:
30 years experience in running (more than 30) projects with industry partners in areas such as: energy; infrastructure; telecommunication; production; logistics, traffic and transport; VLSI design
Martin Grötschel
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Network, Line and Fare Planning (Potsdam)
Optimization
Overview
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Telecommunication Network Design
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Logical connections: solution Physical connections: solution
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Optimization and control of transport devices (such as elevators, stacker cranes) in factories
Herlitz, Falkensee
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Optimization of Botany Bay a Container Terminal in Sydney, Australia
Martin Grötschel
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Gary Froyland, Thorsten Koch, Nicole Megow, Emily Duane, Howard Wren: Optimizing the Landside Operation of a Container Terminal, OR Spektrum 2007
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Proceedings of a meeting of the DMV, German Mathematical Society,
1993
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acatech German National Academy of Engeneering
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Preface Production Factor Mathematics
Martin Grötschel
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Preface Production Factor Mathematics
Martin Grötschel
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My lecture: Basics of
Modelling
Simulation
Optimization
to achieve a joint understanding of the terms and to agree what we mean with these „key words“.
Martin Grötschel
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Martin Grötschel
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Modelling: What is that? http://3.bp.blogspot.com/_vPvIQI8sCvI/TEOtnHyl6xI/AAAAAAAAAD8/T7QhnpW-T1o/s1600/godfroy_harvard_architecture_school_model.jpg
Three models of a water molecule (polar). From left: stick (structural) model, Bohr model, space-filling (cloud) model.
http://www.lionden.com/chemistry_models.htm
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Modelling: What is that?
Olga Koklowa Picasso‘s first wife
„Art is a lie that makes us realize truth“
A mathematical model is a lie that helps us see the truth.
The model of the artist
The model of the mathematician
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Picasso and his women: abstraction
Olga Francesca
Marie-Thérèse
Jacqueline Francoise
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Martin Grötschel
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Mathematical Modelling: What is that? Beginning with observations
of our environment
a problem in practice of particular interest or
a physical, chemical, or biological phenomenon
and with guiding/tailored experiments:
the attempt of a formal representation via „mathematical concepts“ (variables, equations, inequalities, objective functions , etc.), aiming at the utilization of mathematical theories and tools.
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Sketch of wave propagation by Heinrich Hertz
Annalen der Physik und Chemie N. F. Bd. XXXVI, 1889
Electromagnetic Waves
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Electromagnetic Waves & James Clerk Maxwell
(1831-1879)
Maxwell equations 1873
rot
rot
t
t
∂= +
∂∂
= −∂
H j D
E B
div 0
div ρ
=
=
BD
+Material equations
Ampère’s law
Faraday’s law
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Martin Grötschel
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Configuration of Antennas in Mobile Telcommunication
path loss
Isotropic Prediction
Antenna Prediction Available for each potential antenna location
Antenna Configuration Azimuth Tilt Height
height: 41m, electrical tilt: 0-8°, azimuth 0-120°
© Digital Building Model Berlin (2002), E-Plus Mobilfunk GmbH & Co. KG
© Digital Building Model Berlin (2002), E-Plus Mobilfunk GmbH & Co. KG, Germany
Antenna Diagram Signal propagation
in different directions
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Antennas & Interference
Martin Grötschel
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x x
antenna
backbone network
x x
x x site
x x
cell
co- & adjacent channel
interference cell
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Region Berlin - Dresden
Martin Grötschel
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2877 carriers
50 channels
Interference reduction: 83.6%
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Mathematical models of integer/combinatorial optimization Maximum flow, min-cost flow and other flow problems shortest paths (with resource constraints) travelling salesman problem (the prototype problem) routing location set-packing, -partitioning, -covering cut problems (max and min) scheduling node and edge colouring general integer and mixed-integer prorams etc.
Martin Grötschel
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Martin Grötschel
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Typical optimization models (problems)
max ( ) min ( )( ) 0, 1,2,...,( ) 0, 1, 2,...,
( )
i
j
n
f x or f xg x i kh x j m
x and x S
= =≤ =
∈ ∈R
min
0
=≤
≥
Tc xAx aBx b
x n
{ }
min
0
( 0,1 )
T
n
n
c xAx aBx b
xx
x
=≤≥
∈
∈
Z
linear program
LP
(linear) integer
program IP, MIP
„general“ (nonlinear) program
NLP
program = optimization problem
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Minimum Interference Frequency Assignment Problem (FAP) FAP viewed as an Integer Linear Program (IP Model):
Martin Grötschel
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{ }
min
. . 1
1 , ( )1 ,1 , 1
, , 0,1
co ad
v
co co ad advw vw vw vw
vw E vw E
vff F
dwgvf
co covw v wvf wfad ad
wg vwvfco advw vwvf
c z c z
s t x v V
x x vw E f g d vwx x z vw E f F Fx x z vw E f gx z z
∈ ∈
∈
+
= ∀ ∈
+ ≤ ∀ ∈ − <
+ ≤ + ∀ ∈ ∈ ∩
+ ≤ + ∀ ∈ − =
∈
∑ ∑∑
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Martin Grötschel
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Martin Grötschel
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Simulation Simulation, Simulator or simulate are derived from the
latin words simulare and similis .
They mean: pretend to be or the same sort.
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Martin Grötschel
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Simulation „Computation“ of several (close to reality) variants of a
mathematical modell aimimg at: „validation“of the correctness of a modell
investigation of typical instances in the modell framework, e.g., to avoid experiments or to test some functionality (crash-test)
good predictions (weather)
computation of reasonble solutions for the control of a system in practice (control of transport and logistics-systems)
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Martin Grötschel
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Simulation Computation of instances varying several parameters
Parameters of a car crash test, e.g.: speed, material stiffness, various angles
3D-reconstruction of a scull from a magneto-resonance tomografic investigation
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Martin Grötschel
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Configuration of Antennas in Mobile Telcommunication
path loss
Isotropic Prediction
Antenna Prediction Available for each potential antenna location
Antenna Configuration Azimuth Tilt Height
height: 41m, electrical tilt: 0-8°, azimuth 0-120°
© Digital Building Model Berlin (2002), E-Plus Mobilfunk GmbH & Co. KG
© Digital Building Model Berlin (2002), E-Plus Mobilfunk GmbH & Co. KG, Germany
Antenna Diagram Signal propagation
in different directions
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Sea level changes 1993-2010
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Tsunami Model Simulation
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What simulation cannot do! In contrast to a frequent belief in enginering and management science:
Simulation can‘t optimize, can’t even find local minima!
Simulation can‘t even decide infeasibility!
Simulation can‘t disprove anything!
Simulation may be able to find a feasible solution, but is not able to prove its quality (near optimality, quality guarantee).
Auction results need to be “court proof”. Simulation can’t provide legally unchallengeable results.
Martin Grötschel
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Satisfiability of Logical Formulas & Chip Design Verification:
A truly important application: Verification von Computer Chips and “Systems on Chips”
A logic design is correct if and only if a system of certain logic formulas is false.
1 2 1 2 3 1 2 1 2 3 1 2( ) ( ) ( ) ( ) ( )¬ ∨ ∧ ∨ ∨ ∧ ∨¬ ∧ ∨ ∨¬ ∧ ¬ ∨¬x x x x x x x x x x x x
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Given: a chip specification in a hardware modelling language
(e.g. VHDL, Verilog)
a property of the registers in the circuit
Question: Is the property valid for all possible register values that satisfy
the constraints of the chip specification?
There may be 21000 states. How can you find that one of these states is wrong (leading to a possibly hazardous chip error) by simulating some states?
Property Checking
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How does one decide whether an existing gas network is good enough for an expected demand?
Check 50 cases?
Property Checking
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Simulation: the positive side Simulation is indispensable, though
To handle situations too complicated to model exactly
To obtain a „good“ understandig of complex situations
To see whether the simplifications are justified
To check solutions of models
Martin Grötschel
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Martin Grötschel
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Martin Grötschel
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Typical optimization models (problems)
max ( ) min ( )( ) 0, 1,2,...,( ) 0, 1, 2,...,
( )
i
j
n
f x or f xg x i kh x j m
x and x S
= =≤ =
∈ ∈R
min
0
=≤
≥
Tc xAx aBx b
x n
{ }
min
0
( 0,1 )
T
n
n
c xAx aBx b
xx
x
=≤≥
∈
∈
Z
linear program
LP
(linear) integer
program IP, MIP
„general“ (nonlinear) program
NLP
program = optimization problem
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MIP Solver Techniques
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(Operations Research, Jan 2002, pp. 3—15, updated in 2004)
Algorithms (machine independent):
Primal versus best of Primal/Dual/Barrier 3,300x
Machines (workstations →PCs): 1,600x
NET: Algorithm × Machine 5,300,000x
(2 months/5300000 ~= 1 second)
Progress in LP: 1988—2004
Courtesy Bob Bixby
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1
10
100
1000
10000
100000
0
1
2
3
4
5
6
7
8
9
10
1.2→2.1 2.1→3 3→4 4→5 5→6 6→6.5 6.5→7.1 7.1→8 8→9 9→10 10→11
Cum
ulat
ive
Spee
dup
Vers
ion-
to-V
ersi
on S
peed
up
CPLEX Version-to-Version Pairs
V-V Speedup Cumulative Speedup
Mature Dual Simplex: 1994
Mined Theoretical Backlog: 1998 29.530x
MIP Speedups 1991 – 2008
1½ years in 1991 ~ 1 second in 2008
min
0
=≤≥
∈
T
n
c xAx aBx b
xx Z
Courtesy Bob Bixby
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ZIB MIP Instances Variables Constraints Non-zeros Description
1 12,471,400 5,887,041 49,877,768 Group Channel Routing on a 3D Grid Graph (Chip-Bus-Routing)
2 37,709,944 9,049,868 146,280,582 Group Channel Routing on a 3D Grid Graph (different model, infeasible)
3 29,128,799 19,731,970 104,422,573 Steiner-Tree-Packing on a 3D Grid Graph
4 37,423 7,433,543 69,004,977 Integrated WLAN Transmitter Selection and Channel Assignment
5 9,253,265 9,808 349,424,637 Duty Scheduling with base constraints
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Gas Transport Optimization
Martin Grötschel
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Matheon Institutions
Zuse-Institut Berlin
Humboldt U
zu Berlin
Weier-straß
Institut
Leibniz U
Hannover
U Duisburg-
Essen
F.-A. U Erlangen-Nürnberg
TU Braun-schweig
Matheon B20
Optimization of Gas Transport
Bundesnetzagentur (BNetzA)
Open Grid Europe (OGE)
BMWi
Technical Capacity of Gas Networks
OGE ForNe
Research Cooperation Network Optimization
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Martin Grötschel
Modelling Aspects of Gas Transportation Nonlinear Nonconvex: • Loss of pressure over pipes:
• Power of compressor:
Mixed-Integer: • Flow direction • Coupling constraints
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Martin Grötschel
Modelling Aspects of Gas Transportation Uncertainty: • Stochastic nomination at exits • Unknown nomination at entries
Constraint Integer Programming: • combines SAT, MIP, and CP strong modeling capability full power of MIP solving techniques
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Martin Grötschel
Aspects of Gas Transportation The OGE problem is a: • Stochastic • Mixed • Integer • Non • Linear • Constraint • Program
It consists of: • Stochastic Part • Mixed Integer Part • Non-Linear Part
• Constraint Integer Programming Part
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92 convex and non-convex MIQCP
Martin Grötschel
50
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The Real Problem
Pure Mathematics Computer Science
Mathematical Model
Mathematical Theory
Design of Good Solution
Algorithms
Algorithmic Implementation
Numerical Solution
Quick Check: Heuristics
Hard- ware
Soft- ware Data GUI
Practitioner Specialist Modelling
Simulation
Optimization
The problem solving cycle in modern applied mathematics
The Application Driven Approach
Simulation
Implementation in Practice
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Martin Grötschel Institut für Mathematik, Technische Universität Berlin (TUB) DFG-Forschungszentrum „Mathematik für Schlüsseltechnologien“ (MATHEON) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)
[email protected] http://www.zib.de/groetschel
From Simulation to Optimization
Martin Grötschel Workshop
MANAGEMENT OF GAS NETWORKS mathematical solution technologies
Berlin, October 15