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Integer Programming
IFOR Events
Seminar:
'Optimization & Applications'
(see information & program)
Feb 20 - May 28, 2012
IFOR Mitteilungen
This booklet informs about ongoing projects and future events at the IFOR and appears once at the end of the year.
| Lecturer |
Prof. Komei Fukuda Dr. Jan Foniok |
Time |
V: Tu 15–17 U: Tu 17–18 |
| Assistants |
Lorenz Klaus |
Place | V + U: HG E1.1 |
Description
(Mixed) Integer programming deals with problems of minimizing or maximizing a linear function of many variables subject to linear inequality constraints and integrality restrictions on (some of) the variables.
The main objective is to learn the basic theory and algorithms for integer programming and mixed integer programming.
The following topics will be covered in the lecture:
- IP/MIP modeling of combinatorial optimization problems,
- Optimality, relaxation and bounds,
- Well-solved IP problems,
- Euclidean algorithm and Hermite normal form,
- basis reduction in lattices,
- complexity and problem reductions,
- branch-and-bound review,
- cutting plane algorithms,
- Lagrangian duality,
- further techniques and applications.
Prerequisites and Remarks
This course assumes the basic knowledge of linear programming, which is taught in courses such as "Optimization Techniques" (401-3901-00L) and "Introduction to Optimization" (401-2903-00L).
The former title of this course unit was "Topics in Discrete Optimization". If you already got credits for "Topics in Discrete Optimization" (401-3902-00L), you cannot get credits for "Integer Programming" (401-3902-00L).
Evaluation
Solving at least 50% of exercise problems is required for a student to qualify for the exam. In order (for both a masters and doctorate student) to obtain credits, one has to pass the oral exam during an ordinary exam period. The duration is thirty minutes.
Lecture notes
An updated version of the lecture notes (1 June 2010). Please email us if you find any further errors.
Schedule
| Date | Topic | Notes | Exercises | Solutions | Other material |
| 23.02.2010 | Introduction to IP/MIP. Modeling. | Chapter 1 | Assignment 1 | ||
| 02.03.2010 | Euclidean Algorithm, Hermite Normal Form | Chapter 2 | Assignment 2 | ||
| 09.03.2010 | Lattices, Fourier-Motzkin elimination | Chapter 3 | Assignment 3 | ||
| 16.03.2010 | Polyhedra, Convexity | Chapter 3 & 4 | Assignment 4 | ||
| 23.03.2010 | Complexity of IP, Hilbert Basis | Assignment 5 | |||
| 30.03.2010 | Complexity of IP, Well Solved Problems | Chapter 4 & 5 | None | ||
| 06.04.2010 | Easter break | ||||
| 13.04.2010 | Diophatine Approximation, Lattice Reduction | Chapter 6 (2nd update) | Assignment 6 |
Mathematica notebook: Shortest Vector |
Mathematica notebooks: Simultaneous Diophantine Approximation Approximate Closest Vector |
| 20.04.2010 | Integer Programming, Lenstra's algorithm | Chapter 7 (1st update) | Assignment 7 | ||
| 27.04.2010 | Integral Polyhedra, Totally Unimodular Matrices | Chapter 8 | Assignment 8 | ||
| 04.05.2010 | Valid Inequalities | Chapter 9 (1st update) | Assignment 9 | ||
| 11.05.2010 | Chvatal-Gomory Inequalities | Chapter 9 (2nd part) | Assignment 10 | ||
| 18.05.2010 | IP Algorithms | Chapter 10 | Assignment 11 | ||
| 25.05.2010 | Lagrangian Duality | Chapter 11 | Assignment 12 | ||
| 01.06.2010 | Branch and Bound | None |
References
For the theory of integer programming, excellent sources are [8,3,10]. Many of our course materials benefit from these books and research papers. Combinatorial optimization problems yields important models of integer programming. Some of the excellent books on combinatorial optimization are [4,5,9]. There are a few classical books, e.g. [6,7] that are still very useful.
There is a very useful free software LP_SOLVE for mixed integer programming [2], while CPLEX is a highly reliable commercial MIP solver. Another useful software related to integer programming is 4TI2 [1] which can be used to compute, for example, Hilbert bases of polyhedral cones. Those taking this course are expected to install both LP_SOLVE and 4TI2.
|
[1] |
4ti2 team. 4ti2 - a software package for algebraic, geometric and combinatorial problems on linear spaces. Available at www.4ti2.de. |
| [2] | M. Berkelaar and J. Dirks. LP_SOLVE, a mixed integer linear programming (MILP) solver. http://lpsolve.sourceforge.net |
| [3] | D. Bertsimas and R. Weismantel. Optimization over integers. Dynamic Ideas, Belmont, 2005. |
|
[4] |
W. Cook, W. Cunningham, W. Pullyblank, and A. Schrijver. Combinatorial optimization. Series in Disctrete Mathematics and Optimization. John Wiley & Sons, 1998. |
| [5] | M. Grötschel, L. Lovász, and A. Schrijver. Geometric algorithms and combinatorial optimization. Springer-Verlag, Berlin, 1988. |
| [6] | E. L. Lawler. Combinatorial optimization: networks and matroids. Holt, Rinehart and Winston, New York, 1976. |
| [7] | C. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Printice-Hall, 1982. |
| [8] | A. Schrijver. Theory of linear and integer programming. John Wiley & Sons, New York, 1986. |
| [9] | A. Schrijver. Combinatorial optimization. Polyhedra and efficiency. Vol. A, B, C, volume 24 of Algorithms and Combinatorics. Springer-Verlag, Berlin, 2003. |
| [10] | L. Wolsey. Integer programming. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, New York, 1998. |
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