2026 (Current Year) Faculty Courses School of Computing Department of Mathematical and Computing Science Graduate major in Mathematical and Computing Science
Mathematical Models and Computer Science
- Academic unit or major
- Graduate major in Mathematical and Computing Science
- Instructor(s)
- Makoto Yamashita / Kenji Amaya / Daisuke Kurabayashi / Hanna Sumita
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- MCS.T506
- Number of credits
- 200
- Course offered
- 2026
- Offered quarter
- 4Q
- Syllabus updated
- Jul 13, 2026
- Language
- Japanese
Syllabus
Course overview and goals
This course studies basic ideas of mathematical optimization, with focus on convex analysis and optimization algorithms. In the first part, students learn basic viewpoints of optimization, separation theorems, convexity, subgradients, and conjugate functions. Then the course covers gradient-based methods for smooth optimization, Nesterov acceleration, Newton method, and also proximal mapping, Moreau envelope, and the proximal point method for nonsmooth convex optimization. For constrained convex optimization, the course covers duality, KKT conditions, sensitivity analysis, linear programming, optimal transport, and the basic idea of interior-point methods. In the later part, students study tangent cones, linearization, constraint qualifications, KKT conditions for nonlinear constraints, and the augmented Lagrangian method together with the dual proximal point method. The course also includes engineering applications and efficient computation in optimization, so that students can learn optimization from a broader viewpoint while seeing the connection between theory and practice.
Recent progress of computers has changed mathematical methodology in a large way. Mathematical optimization is one important tool in this change. It is important to understand both theory and algorithms, so that students can model problems correctly and choose suitable numerical methods. This course studies optimization from both geometric viewpoints and algorithmic viewpoints, and also keeps connections to applications.
Course description and aims
At the end of this course, students will be able to:
1. Explain basic concepts in convex optimization, such as convexity, separation theorems, subgradients, and conjugate functions.
2. Explain the framework of basic numerical methods, including gradient methods, Nesterov acceleration, Newton method, and the proximal point method.
3. Explain the basic ideas of duality, KKT conditions, and interior-point methods.
4. Explain the relations among tangent cones, linearization, constraint qualifications, and KKT conditions for nonlinear constraints.
5. Explain the relation between the augmented Lagrangian method and the dual proximal point method, and understand how these ideas connect to engineering applications and efficiency improvements.
Keywords
Mathematical optimization, convex analysis, separation theorem, supporting hyperplane, subgradient, conjugate function, dual norm, gradient method, Nesterov acceleration, Newton method, proximal mapping, Moreau envelope, duality, KKT conditions, interior-point method, optimal transport, tangent cone, augmented Lagrangian method
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Classes are given mainly by lecture slides. At the end of each class, short exercises or review problems are given. When needed, simple examples and calculations are used, so that students can check the meaning of definitions, theorems, and algorithms.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Introduction, Optimization, Separation, Convexity |
Understand basic examples of optimization, and explain the basic ideas of epigraphs and convexity. |
| Class 2 | Separation, Supporting Hyperplanes, Subgradients |
Explain how supporting hyperplanes and subgradients are obtained from separation theorems. |
| Class 3 | Conjugates, Fenchel-Young Inequality, Dual Norms |
Explain the meaning of conjugate functions, the Fenchel-Young inequality, and dual norms. |
| Class 4 | Smooth Unconstrained Optimization - Armijo Condition and Backtracking |
Explain how step sizes are chosen by the Armijo condition and backtracking line search. |
| Class 5 | Complexity - Gradient Descent and Nesterov Acceleration |
Explain the difference in convergence behavior between gradient descent and Nesterov acceleration. |
| Class 6 | Newton Method - Local Quadratic Convergence |
Explain the basic idea of Newton method and local quadratic convergence. |
| Class 7 | Nonsmooth Convex Optimization - Proximal Mapping, Moreau Envelope, PPM |
Explain the basic ideas of proximal mapping, the Moreau envelope, and the proximal point method. |
| Class 8 | Convex Constraints - Examples, Duality, KKT Conditions, Sensitivity |
Explain the basic framework of Lagrangian duality, KKT conditions, and sensitivity analysis. |
| Class 9 | LP and Optimal Transport - Farkas Lemma, Duality, Complementarity, Interior-Point |
Explain the basic ideas of LP duality, complementarity, and interior-point methods, and view optimal transport as a linear programming problem. |
| Class 10 | Optimization in Mechanical Measurement |
Explain what kind of problems in mechanical measurements are solved by optimization. |
| Class 11 | Optimization Methods that Imitate Nature |
Explain the relation between optimization problems around us and nature-inspired optimization methods. |
| Class 12 | Efficiency Improvements on Optimization Methods |
Understand various improvement methods for optimization algorithms. |
| Class 13 | Nonlinear Constraints - Tangent Cones, Linearization, Constraint Qualifications, KKT |
Explain the relations among tangent cones, linearized cones, constraint qualifications, and KKT conditions. |
| Class 14 | Augmented Lagrangian Method (ALM), Dual PPM, and Wrap-up |
Explain the basic idea of the augmented Lagrangian method and its relation to the dual proximal point method. |
Study advice (preparation and review)
To improve learning results, students should use lecture notes and reference books, and spend about 100 minutes before each class for preparation and about 100 minutes after each class for review, including assignments.
Textbook(s)
No textbook is required. The course is mainly based on lecture materials.
Reference books, course materials, etc.
The main references are as follows.
S. Boyd and L. Vandenberghe, "Convex Optimization", Cambridge University Press, 2004.
J. Nocedal and S. Wright, "Numerical Optimization (2nd ed.)", Springer, 2006.
R. T. Rockafellar, "Convex Analysis", Princeton University Press, 1970.
D. P. Bertsekas, "Nonlinear Programming (3rd ed.)", Athena Scientific, 2016.
Other references may be introduced in the lecture materials when needed.
Evaluation methods and criteria
Students are evaluated on their understanding of mathematical modeling by optimization, the theory in the course, and the framework of the main numerical methods. The final grade is based on one report and a final exam.
Related courses
- MCS.T302 ： Mathematical Optimization
- MCS.T402 ： Mathematical Optimization: Theory and Algorithms
- ICT.M310 ： Mathematical Programming
- IEE.A430 ： Numerical Optimization
Prerequisites
The following knowledge are required.
* The simplex method for linear programming problems
* Linear algebra (in particular, positive semidefinite matrices)