Martin Grötschel Institut für Mathematik, Technische Universität Berlin (TUB) DFG-Forschungszentrum „Mathematik für Schlüsseltechnologien“ (MATHEON) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)

groetschel@zib.de 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

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

Optimization and control of transport devices (such as elevators, stacker cranes) in factories

Herlitz, Falkensee

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

Proceedings of a meeting of the DMV, German Mathematical Society,

1993

acatech German National Academy of Engeneering

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Preface Production Factor Mathematics

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Preface Production Factor Mathematics

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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“.

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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

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

Picasso and his women: abstraction

Olga Francesca

Marie-Thérèse

Jacqueline Francoise

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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.

Sketch of wave propagation by Heinrich Hertz

Annalen der Physik und Chemie N. F. Bd. XXXVI, 1889

Electromagnetic Waves

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

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

Antennas & Interference

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x x

antenna

backbone network

x x

x x site

x x

cell

co- & adjacent channel

interference cell

Region Berlin - Dresden

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2877 carriers

50 channels

Interference reduction: 83.6%

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

Minimum Interference Frequency Assignment Problem (FAP) FAP viewed as an Integer Linear Program (IP Model):

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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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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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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)

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

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

Sea level changes 1993-2010

Tsunami Model Simulation

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

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

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

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

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

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

Martin Grötschel

Modelling Aspects of Gas Transportation Nonlinear Nonconvex: • Loss of pressure over pipes:

• Power of compressor:

Mixed-Integer: • Flow direction • Coupling constraints

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

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

92 convex and non-convex MIQCP

Martin Grötschel

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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

Martin Grötschel Institut für Mathematik, Technische Universität Berlin (TUB) DFG-Forschungszentrum „Mathematik für Schlüsseltechnologien“ (MATHEON) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)

groetschel@zib.de http://www.zib.de/groetschel

From Simulation to Optimization

Martin Grötschel Workshop

MANAGEMENT OF GAS NETWORKS mathematical solution technologies

Berlin, October 15