linear and integer programming (adm ii) script rolf...

Linear and Integer Programming(ADM II)

Script

Rolf MöhringWS 2010/11

Contents

1-1

1. Introduction................................................................................................................................ 31.1 Algorithmic Discrete Mathematics (ADM)

.................................................................................................................................................................... 41.2 Content of ADM II................................................................................................................................................................. 51.3 Winter term 2010/11

2. Optimization problems................................................................................................................................................................................. 72.1 Examples

...................................................................................................................................... 82.2 Neighborhoods and local optimization...................................................................................................................................................... 92.3 Convex sets and functions

............................................................................................................................................... 102.4 Convex optimization problems3. The Simplex algorithmn

....................................................................................................................................................... 123.1 Forms of linear programs......................................................................................................................................................... 133.2 Basic feasible solutions

........................................................................................................................................... 143.3 The geometry of linear programs........................................................................................................................ 153.4 Local search among basic feasible solutions

......................................................................................................................................................... 163.5 Organisation in tableaus................................................................................................................................................. 173.6 Choosing a profitable column

...................................................................................................................................................... 183.7 Pivoting rules and cycling........................................................................................................................................... 193.8 Phase I of the simplex algorithm

............................................................................................................................................... 203.9 Geometric aspects of pivoting4. Duality

.................................................................................................................................. 224.1 Duality of LPs and the duality theorem..................................................................................................................................................... 234.2 Complementary slackness

.................................................................................................................................. 244.3 The shortest path problem and its dual....................................................................................................................................................................... 254.4 Farkas' Lemma

............................................................................................................................................... 264.5 Dual information in the tableau.................................................................................................................................................. 274.6 The dual Simplex algorithmn

5. Computational aspects of the Simplex algorithm............................................................................................................................................... 295.1 The revised simplex algorithm

................................................................................................. 305.2 Algorithmic consequences of the revised simplex algorithm................................................... 315.3 Solving the max-flow problem with the revised simplex algorithm and column generation

Contents

1-2

.......................................................................................................... 325.4 The simplex algorithmus with lower and upper bounds..................................................................................................................... 335.5 A special case: the network simplex algorithm

6. Primal-dual algorithms............................................................................................................................................................................. 356.1 Introduction

..................................................................................................................................................... 366.2 The primal-dual algorithmn.................................................................................................................................. 376.3 Remarks on the primal-dual algorithmn

......................................................................................................... 386.4 A primal-dual algorithmn for the shortest path problem........................................................................................................ 396.5 A primal-dual algorithmn for the transportation problem

............................................................................... 406.6 A primal-dual algorithmn for the weighted matching problem (a sketch)7. Integer linear optimization

............................................................................................................................................................................. 427.1 Introduction................................................................................................................................................... 437.2 Totally unimodular matrices................................................................................................................................................. 447.3 Branch and bound algorithms

............................................................................................................................................................. 457.4 Lagrangian relaxation......................................................................................................................................................... 467.5 Cutting plane algorithms

................................................................................................................................................... 477.6 Optimization and separation8. Polytopes induced by combinatorial optimization problems

............................................................................................................................................................................. 498.1 Introduction........................................................................................................................................................ 508.2 Some linear descriptions

.................................................................................................................................................... 518.3 Separtion and branch & cut9. LP-based approximation algorithms

...................................................................................................................... 539.1 Simple rounding and the use of dual solutions............................................................................................................................................................. 549.2 Randomized rounding

.................................................................................................... 559.3 Primal-dual approximation algorithms and network design10. Complexity of linear optimization and interior point methods

.............................................................................................................................................................. 5710.1 LP is in NP ∩ coNP......................................................................................................................................... 5810.2 Runtime of the simplex algorithm

............................................................................................................................................................ 5910.3 The ellipsoid method........................................................................................................................................................... 6010.4 Interior point methods

1. Introduction

2

................................................................................................................................... 31.1 Algorithmic Discrete Mathematics (ADM)........................................................................................................................................................................ 41.2 Content of ADM II

.................................................................................................................................................................... 51.3 Winter term 2010/11

1. Introduction1.1 Algorithmic Discrete Mathematics (ADM)

3-1

On the history of ADM

Young area, has its roots in

algebra, graph theory, combinatorics

computer science (algorithm design and complexity theory)

optimization

Deals with optimization problems having a disrete structure

graphs, networks

finite solution space

Applications

telecommunication networks, traffic networks

logistics, production planning, location planning

...

ADM at TU Berlin

1. Introduction1.1 Algorithmic Discrete Mathematics (ADM)

3-2

Basic courses

Graph and network algorithms (ADM I)

Linear and integer optimization (ADM II)

Special courses (ADM III)

Scheduling problems

Applied network optimization

Polyhedral theory

...

Seminar (partly parallel with ADM II or ADM III)

Bachelor thesis or master thesis

1. Introduction1.2 Content of ADM II

4-1

Linear optimization problems

Linear objective function, linear inequalities as side constraints

Linear optimization: min cTx subject to Ax ! b, x " 0

Simplex algorithm

Duality

Geometry of linear optimization problems

Ax ! b, x " 0 define a polyhedron


4-2

Optimum is attained in a vertex (corner point)

The simplex algorithm traverses vertices


4-3

Discrete problems as linear optimization problems

polyhedral theory

Discrete problems as geometric problems

Minimum spanning trees as vectors

Graph G


4-4

1 2

3

Minimum spanning trees of G as vectors (incidence vectors)

2

3

1 2 1

3110

!101

!011

!

Convex hull of incidence vectors = polytope (yellow set)

polytope = yellow set

Computing a minimum spanning tree = linear optimization over this polytope


4-5

Integer linear optimization

variables may only attain integer values

much more difficult problems

Solution methods

Lagrangian relaxation

cutting plane algorithms

LP-based approximation algorithms

...

Exercises with implementation assignments

1. Introduction1.3 Winter term 2010/11

5-1

Torsten Gellert (Exercises)

Christoph Hansknecht (Tutorials)

Website

http://www.math.tu-berlin.de/coga/teaching/wt08/adm2/

http://www.math.tu-berlin.de/coga/teaching/wt10/adm2/

Notebook: http://www.math.tu-berlin.de/~moehring/adm2/

Literature

C.#H. Papadimitriou and K.#Steiglitz

Combinatorial Optimization: Algorithms and Complexity

Prentice Hall, Englewood Cliffs, NJ, 1982

Pocket book - 512 pages - Dover Publications

First published: Juli 1998

Auflage: Unabridged

ISBN: 0486402584

B. Korte, J. Vygen:

Combinatorial Optimization: Theory and Algorithms

Springer, 2000/2002/2006/2008


5-2

Springer, 2000/2002/2006/2008

jetzt auch auf deutsch

W. J. Cook, W. H. Cunningham, W. R. Pulleyblank and A. Schrijver

Combinatorial Optimization

Wiley 1998

V.#Chvátal

Linear Programming

Freeman, New York, 1983

G.#L. Nemhauser and L.#A. Wolsey

Integer and Combinatorial Optimization

John Wiley & Sons, New#York, 1988

M.#Grötschel, L.#Lovász, and A.#Schrijver

Geometric Algorithms and Combinatorial Optimization

Springer-Verlag, Berlin, 2nd#ed., 1993

D.#S. Hochbaum, ed.

Approximation Algorithms for NP-hard Problems

PWS Publishing Company, Boston, MA, 1997


5-3

PWS Publishing Company, Boston, MA, 1997

H.#M. Salkin and K.#Mathur

Foundations of Integer Programming

North-Holland, Amsterdam, 1989.

R.#J. Vanderbei

Linear Programming: Foundations and Extensions

Kluwer Acad. Publ., Dordrecht, 2nd#ed., 2001.

http://www.princeton.edu/~rvdb/LPbook/index.html

Encyclopedia

A. Schrijver:

Combinatorial Optimization: Polyhedra and Efficiency

Springer, 2003

3 volumes with 1881 Seiten, aso available as CD

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