2022 Theory and Application of Discrete Optimization

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Academic unit or major
Graduate major in Industrial Engineering and Economics
Shioura Akiyoshi 
Class Format
Lecture    (Face-to-face)
Media-enhanced courses
Day/Period(Room No.)
Tue3-4(W9-508)  Fri3-4(W9-508)  
Course number
Academic year
Offered quarter
Syllabus updated
Lecture notes updated
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Course description and aims

We discuss an auction model with many indivisible (discrete) goods. It is known that an optimal allocation of goods as well as equilibrium prices can be computed by algorithms (protocols) called iterative auctions. In this lecture, we review various iterative auctions and investigate them from the viewpoint of discrete optimization. In particular, we explain the concept of gross-substitutes valuation, which plays a crucial role in the auction, and show the connectin with discrete concavity.

This lecture aims to enable students to understand the power of theoretical results in discrete optimization in application to auction theory in economics.

Student learning outcomes

By the end of this course, students will be able to do the following:
(1) explain the auction model with indivisible goods,
(2) understand the concept of gross-substitutes condition and its properties,
(3) explain how iterative auctions find equilbrium prices,
(4) understand the connection between iterative auctions and optimization algorithms.


auction, discrete optimization, equilibrium, algorithm

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

In each class the instructor gives a lecture. At the end of the lecture, the instructor presents some problems for exercise.

Course schedule/Required learning

  Course schedule Required learning
Class 1 overview of the lecture Details will be given in each lecture.
Class 2 Review of shortest path problem (1)
Class 3 Review of shortest path problem (2)
Class 4 Matching problem on a bipartite graph (1)
Class 5 Matching problem on a bipartite graph (2)
Class 6 optimality condition for maximum matching
Class 7 Maximum-weight matching problem
Class 8 relationship between maximum-weight matching and equilibrium allocation
Class 9 algorithm for computing an equilibrium approximately
Class 10 algorithm for computing an equilibrium exactly
Class 11 multi-demand model and equilibrium
Class 12 gross substitutes property for valuation functions
Class 13 algorithm for computing an equilibrium approximately
Class 14 algorithm for computing an equilibrium exactly

Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.



Reference books, course materials, etc.

Related Paper:
K. Murota, A. Shioura, and Z. Yang: Time bounds for iterative auctions: a unified approach by discrete convex analysis, Technical Report METR 2014-39, University of Tokyo, December 2014.

Related Book:
K. Murota: Discrete Convex Analysis, SIAM, 2003

Assessment criteria and methods

Evaluation based on reports and exams

Related courses

  • IEE.A206 : Operations Research
  • IEE.A330 : Advanced Operations Research

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Knowledge about the theory of combinatorial optimization is required.
In particular, students should have knowledge about the optimality conditions and algorithms
of the minimum spanning tree problem, the maximum (weight) matching problem, and the network flow problem.

Office hours

Any time. Prior appointment by e-mail is required.

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