Mathematical Optimization: Theory and Algorithms

Academic unit or major

Graduate major in Mathematical and Computing Science

Instructor(s)

Mituhiro Fukuda / Makoto Yamashita

Class Format

Lecture (Zoom)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

3-4 Mon (W834) / 3-4 Thu (W834)

Class

-

Course Code

MCS.T402

Number of credits

200

Course offered

2020

Offered quarter

2Q

Syllabus updated

Jul 10, 2025

Language

English

Syllabus

Course overview and goals

This course will cover basic notions to comprehend the gradient-based methods for convex optimization problems considered in mathematical optimization, machine learning and image processing. It starts with the basics, from the definition of convex sets in convex optimization, and will gradually focus on continuously differentiable convex functions. Along the lectures, it will also cover the characterization of solutions of optimization problems (optimality conditions), and numerical methods for general problems such as steepest descent methods, Newton method, conjugate gradient methods, and quasi-Newton methods. In the latter part, the accelerated gradient method of Nesterov for Lipschitz continuous differentiable convex functions will be detailed.

Course description and aims

Objectives: Learn the mathematical concepts and notions from the basics necessary for numerical methods for convex optimization problems.　Definitions and proofs of theorems will be carefully explained. The objective is to understand the role of basic theorems of convex optimization in scientific articles, and to be prepared to apply them for other problems in mathematical optimization and machine learning.
Theme: In the first part, important theorems to analyze convex optimization problems will be introduced. In the second part, the Nesterov's accelerated gradient method which has received a lot of attention in the recent years will be explained from the mathematical point of view.

Keywords

Convex function, algorithm analysis, convex optimization problem, numerical methods in optimization, differentiable convex functions with Lipschitz continuous gradients, accelerated gradient method

Competencies

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

Class flow

All the lectures will have proofs of theorem and explanations of concepts behind a method or definition.

Course schedule/Objectives

Study advice (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.

Textbook(s)

None.

Reference books, course materials, etc.

D. P. Bertsekas, Nonlinear Programming, 2nd edition, (Athena Scientific, Belmont, Massachusetts, 2003).
Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, (Kluwer Academic Publishers, Boston, 2004).
Y. Nesterov, Lectures on Convex Optimization, 2nd edition, (Springer, Cham, Switzerland, 2018).
J. Nodedal and S. J. Wright, Numerical Optimization, 2nd edition, (Springer, New York, 2006).

Evaluation methods and criteria

Understand the basic theorems related to convex sets and convex functions, and the basic numerical methods to solve mathematical optimization problems. Grade will be based on mid-term and final exams or on reports along the course.

Related courses

• MCS.T506 ： Mathematical Models and Computer Science
• IEE.A430 ： Numerical Optimization

Prerequisites

It is necessary that the attendees have basic notions of mathematics such as linear algebra and calculus as well as understand "basic" methodologies to prove.