Many of mathematical problems arising in the areas of economics and industrial engineering can be formulated as discrete optimization problems. In this lecture, we pick up integer programming problems, network programming problems, and combinatorial optimization problems. We explain the mathematical structures of the solution sets of the problems, and connection with combinatorics. We also review various algorithms for discrete optimization problems.
The objective is to let students learn skills which is necessary to solve discrete optimization problems.
By the end of this course, students will be able to:
・Understand fundamental properties of discrete optimization problems.
・Understand fundamental properties of the branch and bound method.
・Understand fundamental properties of typical approximation algorithms.
・Understand fundamental properties of typical heuristics.
discrete optimization problem, integer programming problem, combinatorial optimization problem, the branch and bound method, approximation algorithms, heuristics.
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
In each class we give a lecture in the first half and then assign some exercise problems in the last half.
Course schedule | Required learning | |
---|---|---|
Class 1 | fundamental properties of discrete optimization problems. | understand the contents of each class |
Class 2 | algorithm and complexity | |
Class 3 | overview of linear programming and the duality theorem | |
Class 4 | relaxation method | |
Class 5 | Lagrange relaxation and subgradient method | |
Class 6 | enumerative method | |
Class 7 | linear relaxation problem | |
Class 8 | branch and bounds method | |
Class 9 | dual simplex method | |
Class 10 | additive lower bounds | |
Class 11 | approximation algorithm | |
Class 12 | randomized algorithm | |
Class 13 | heuristics | |
Class 14 | introduction to complexity theory | |
Class 15 | summary |
None required
Course materials will be provided as needed.
Evaluation based on reports and exams
No prerequisites are necessary, but enrollment in related courses is desirable.