combinatorial optimization - ma4502 · pdf filecombinatorial optimization ma4502 ... basics of...

Combinatorial OptimizationMA4502
Michael Ritter
April 14, 2015
Combinatorial Optimization April 14, 2015

Part I
Organisation

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Welcome!

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Who am I?

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Who are you?

Who are you?

Who are you?Are you studying . . .
A Mathematics?B something else?

Who are you?Are you enrolled in . . .
A a masters program?B a bachelors program?

Who are you?Which masters program?
A BiomathematicsB Financial Mathematics and Actuarial SciencesC MathematicsD Mathematics in Operations ResearchE Mathematics in Science and Engineering

Who are you?Did you get your bachelors degree . . .
A at TUM?B somewhere else?

Who are you?Did you attend the course Algorithmische Diskrete Mathematik?
YESNO

Who are you?Did you attend the course Fundamentals of Convex Optimization?
YESNO

Who are you?Did you attend the course Discrete Optimization?
YESNO

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WW What is this?

What is this?optimization problems with combinatorial structuremostly hard instanceslots of LP/ILP techniques

Overview1. Introduction: Notation, Recap,
Typical Problems, Basics of Complexity Theory2. Polyhedral Combinatorics:
valid inequalities, cutting planes, separation,strengthening inequalities
3. Column Generation4. Decomposition5. Approximation Algorithms6. Branch and Bound
7. Optional: Randomized Algorithms

Overview1. Introduction: Notation, Recap,
Typical Problems, Basics of Complexity Theory2. Polyhedral Combinatorics:
valid inequalities, cutting planes, separation,strengthening inequalities
3. Column Generation4. Decomposition5. Approximation Algorithms6. Branch and Bound7. Optional: Randomized Algorithms

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What do I expect?working knowledge of Convex Optimization:
polyhedra
duality theory
simplex algorithm
basics in Algorithmic Combinatorial Optimization:graphs, digraphs
spanning trees
shortest paths
matching
network flow
dynamic programming
independence systems

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What can youexpect?

What can you expect?In General:
focused on applicationsnice theoryslide/blackboard presentationsexercises and interactive elementstutorials: supervised work in groupshome study materials5 credits points

What can you expect?Home Study Materials:
lecture notes before the lectureadditional blackboard presentationshome study sheetstutorial materials available afterwardsstudents exam problems

What can you expect?Tutorials:
recap and students exam problem (25%)discuss homework (25%)supervised group work

What can you expect?Students Exam Problem:
prepare a typical exam problem for the tutorialassess diculty, time needed, creditsprepare write-up and presentationoptional: solution
group of 2-3 studentsassigned 2 weeks in advance

What can you expect?Students Exam Problem:
prepare a typical exam problem for the tutorialassess diculty, time needed, creditsprepare write-up and presentationoptional: solutiongroup of 2-3 studentsassigned 2 weeks in advance

What can you expect?Exam:
60 minutesclosed bookno access restrictions, no bonusone students exam problem

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Why take thiscourse?

Why take this course?You will . . .
learn to solve real-world problemsknow tools and techniquesknow how to analyze combinatorial polyhedrabe able to apply exact solution techniquesbe able to apply approximation techniquesknow how to choose your approach

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What should youdo next?

What next?register on TUM-Online (mailing list)
tutorial registration: today 20:00 until Mondaymid May: exam registrationrecap convex/linear optimizationalgorithmic discrete mathematicsdecide on starting time: 16:00 or 16:15?decide on exam date: July 23, 24, 27, 28

What next?register on TUM-Online (mailing list)tutorial registration: today 20:00 until Monday
mid May: exam registrationrecap convex/linear optimizationalgorithmic discrete mathematicsdecide on starting time: 16:00 or 16:15?decide on exam date: July 23, 24, 27, 28

What next?register on TUM-Online (mailing list)tutorial registration: today 20:00 until Mondaymid May: exam registration
recap convex/linear optimizationalgorithmic discrete mathematicsdecide on starting time: 16:00 or 16:15?decide on exam date: July 23, 24, 27, 28

What next?register on TUM-Online (mailing list)tutorial registration: today 20:00 until Mondaymid May: exam registrationrecap convex/linear optimizationalgorithmic discrete mathematics
decide on starting time: 16:00 or 16:15?decide on exam date: July 23, 24, 27, 28

What next?register on TUM-Online (mailing list)tutorial registration: today 20:00 until Mondaymid May: exam registrationrecap convex/linear optimizationalgorithmic discrete mathematicsdecide on starting time: 16:00 or 16:15?
decide on exam date: July 23, 24, 27, 28

What next?register on TUM-Online (mailing list)tutorial registration: today 20:00 until Mondaymid May: exam registrationrecap convex/linear optimizationalgorithmic discrete mathematicsdecide on starting time: 16:00 or 16:15?decide on exam date: July 23, 24, 27, 28

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What questionsdo you have?

Part II
Introduction and Prerequisites

Recap: Linear & Integer Linear Programs

Integer Linear Programs
max cT xAx bx 0
x Zn
P := {x Rn : Ax b, x 0}P(A, b) := {x Rn : Ax b}
I(P) := conv(P fl Zn)
x1
x2
P
c

max cT xAx bx 0
x Zn
x1
x2
P
c
x

max cT xAx bx 0x Zn
x1
x2
P
c

max cT xAx bx 0x Zn
P := {x Rn : Ax b, x 0}P(A, b) := {x Rn : Ax b}I(P) := conv(P fl Zn) x1
x2
P
c
I(P)

Recap: Polyhedra, Dimension, Faces and Facets

PolyhedraDefinition (polyhedra, polytopes)
intersection of finitely many halfspacesP = {x Rn : Ax b}integer / rational A and b rational polyhedronbounded polyhedron polytope
Definition (dimension)dim(X ) := dim a(X )dim(X ) = n full-dimensional
Co

combinatorial optimization - ma4502 · pdf filecombinatorial optimization ma4502 ... basics of...

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