Lectures: Prof. Dr. Moritz Diehl (email),
Exercises: Léo Simpson (email)
The course provides an introduction to numerical methods for solving optimization problems in science and engineering. The focus is on continuous nonlinear optimization in finite dimensions, covering both convex and nonconvex problems. It is intended for a mixed audience of students from mathematics, engineering, and computer science.
The course is organized as an inverted classroom and based on four pillars:
- a course manuscript
- lecture recordings, and
- exercises that are accompanied by solutions and solution recordings, and
- weekly alternating Q&A and exercise sessions to discuss the course contents
Contact: moritz.diehl@imtek.uni-freiburg.de, leo.simpson@imtek.uni-freiburg.de
*** This page is intended for the course as taught in the summer semester 2025 at the University of Freiburg. For a timeless version with a focus on self-study of the material, see here.***
Organization of the course
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Exam date: XX.XX.2025, XX:00 - XX:00 | Rooms: XXX
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The course is organized as a flipped classroom. We provide recordings of the lecture and will meet once a week to discuss the course contents. This course has 6 ECTS credits. It is possible to undertake a project to earn an additional 3 ECTS, for a total of 9 ECTS for the course and project. For more information, please get in touch with Léo Simpson.
Meetings: We will meet every Tuesday, 14:10 to 15:50 in Room HS II, Albertstraße 23b.
These meetings are alternatingly dedicated to either Q&A sessions with Prof. Diehl or exercise sessions with the teaching assistant (see calendar below), and will not be recorded.
Ilias: Even though most of the material will be published on this page, there is also an Ilias course. Official communication will be conducted through the forum on Ilias. You can also use the forum for discussions related to the course, to ask questions (content-related or organization-related). Please feel free to open new topics and to answer questions of your fellow students. Furthermore, a mid-term quiz will be published on Ilias (see section about the midterm quiz below).
Lecture recordings: The lecture recordings were already created in a past semester. There are 24 lectures, each approximately 90 minutes long. You can find the recordings below in the materials section, and a recommended schedule in the calendar.
Course manuscript: The lectures are accompanied by a detailed course manuscript course. We will provide printed versions.
Exercises: The exercises are mainly computer-based. Computers with Python and CasADi installed are required to solve them (see below for details). The exercises are voluntary, but we strongly recommend that you solve them. We offer the possibility to hand them in to receive feedback, but for this, please respect the deadlines you can find in the calendar below. If you would like feedback on a specific part of the exercise, you can state so on your solution sheet. To hand them in, send them in an email to Léo Simpson.
Q&A sessions: Every second week, there will be a virtual/(in person?) Q&A session with Prof. Diehl, where you can ask any questions about the course content. The format is meant to be highly interactive and depends strongly on your participation. We would recommend that while watching the video lectures or reading the course script, you write down any questions that come to your mind, such that you have them readily available for the Q&A sessions.
Exercise sessions: Every other week, we will meet for the exercise sessions. They will be used to discuss the solutions and any questions related to the exercises. These can either be questions about the current exercise sheet or questions about the solution to the last sheet. As the Q&A sessions, this format depends heavily on your participation.
Mid term quiz: At the middle of the semester, we will publish a quiz on Ilias, with questions covering the course contents so far. It is obligatory that you pass this quiz until 06.06.2025 (23:59). Note that you have infinitely many trials for doing so and will receive instant feedback by auto-grading. The quiz will be online at least one week before the deadline. Note that the questions will not necessarily be representative of an exam.
Final evaluation: The final exam is a written exam. Only pen, paper, a non-programmable calculator and two A4 cheat-sheets (i.e., 4 pages) of handwritten content are allowed. For students from M.Sc. MSE / ESE and B.Sc. Math, this exam is graded. The students from the M.Sc. of mathematics need to pass the written exam in order to take the graded 11ECTS oral exam. Everyone who wants ECTS for this course needs to pass the exam.
Projects: (more detail in a section below) The optional project (3 ECTS) consists in the formulation and implementation of a self-chosen optimization problem or numerical solution method, resulting in documented computer code, a project report, and a public presentation. Project work starts in the last third of the semester. For students from the faculty of engineering the project is graded independently from the 6ECTS lecture. For students from the B.Sc. Math, the grade for the lecture and project 9ECTS module is solely determined by the written exam. For students from the M.Sc. Math the project is again a prerequisite to the graded 11ECTS oral exam.
Calendar
| Date | Format | Content | Watch this week | Read this week | Prepare | Deadlines |
| 22.04 | Introduction | Lec. 1, 2 | Chap. 1 and 2 pages 6 - 19 | one question | ||
| 29.04 | Exercise | Solution to Ex 1 | Lec. 3, 4 | Chap. 3 and 4 pages 20-35 | one question | Ex 1 (voluntary) |
| 06.05 | Q&A | up to Chap. 6 (incl.) | Lec. 5, 6 | Chap. 5 and 6 pages 37-50 | one question | |
| 13.05 | Exercise | Solution to Ex 2 | Lec. 7, 8, 9 | Chap. 7 and 8 pages 51-64 | one question | Ex 2 (voluntary) |
| 20.05 | Q&A | up to Chap. 9 (incl.) | Lec. 10, 11 | Chap. 9 pages 65-73 | one question | |
| 27.05 | Exercise | Solution to Ex 3 | Lec. 12, 13, 14 | Chap. 10 and 11 pages 74-92 | one question | Ex 3 (voluntary) |
| 03.06 | Q&A online? | up to Chap. 13 (incl.) | Lec. 15, 16, 17 | Chap. 12 pages 94-104 | one question | Midterm quiz |
| 09.06-13.06 | *** Lecture break *** | |||||
| 17.06 | Exercise | Solution to Ex 4 | Lec. 18, 19 | Chap. 13 pages 106-118 | one question | Ex 4 (voluntary) |
| 24.06 | Q&A | up to Chap. 14 (incl.) and projects | Lec. 20, 21 | Chap. 14 pages 119-126 | one question | |
| 01.07 | Exercise | Solution to Ex 5 | Lec. 21, 22 | Chap. 15 pages 127-133 | one question | Ex 5 (voluntary) + project commitments |
| 08.07 | Q&A | All course content and projects | Lec. 23, 24 | one question | ||
| 15.07 | Project discussion | Project | project question | |||
| 22.07 | Project presentation | Project presentation | Project presentation | |||
| 05.08 | (no session - just a deadline) | project report | project report deadline | |||
Manuscript
The lectures closely follow a course manuscript draft.
Numerical optimization by Prof. Dr. Moritz Diehl
Lectures
The video recordings correspond to approximately 90 minutes each and comprise 24 lectures in total. A recommended schedule for watching can be found in the calendar above.
| Lecture | Content |
| Lecture 1 | Introduction to Section 1.3 (Mathematical formulation) |
| Lecture 2 | Section 1.4 (Definitions) to 2.7 (Mixed-Integer-Programming) |
| Lecture 3 | Section 3.1 (How to check convexity) to 3.5 (Standard form of convex opt. problems) |
| Lecture 4 | Section 3.6 (Semidefinite Programming) to Example 4.2 (Dual of LP) |
| Lecture 5 | Example 4.3 (Dual decomposition) to Chapter 6 introduction |
| Lecture 6 | Section 6.1 (Linear Least Squares) to 6.5 (L1-Estimation) |
| Lecture 7 | Section 6.6 (Gauss-Newton method) to 7.2 (Local convergence rates) |
| Lecture 8 | Section 7.3 (Newton-Type methods) to 8.1 (Local contraction) |
| Lecture 9 | Section 8.2 (Affine invariance) to 9.1 (Line search) |
| Lecture 10 | Section 9.2 (Wolfe conditions) to 9.3 (Global convergence of line search) |
| Lecture 11 | Section 9.4 (Trust-Region methods) to 9.5 (The Cauchi-Point) |
| Lecture 12 | Section 10.1 (Algorithmic Differentiation) to 10.3 (Backward AD) |
| Lecture 13 | Section 10.4 to Chapter 11 introduction |
| Lecture 14 | Section 11.1 (LICQ and linearized feasible cone) to 11.2 (SONC) |
| Lecture 15 | Section 12.1 (Optimality conditions) to 12.5 (Constrained Gauss-Newton) |
| Lecture 16 | Section 11.3 (Perturbation analysis) and 12.7 (Local convergence) |
| Lecture 17 | Section 12.6 (General constrained NT-Algorithm) to 12.9 (Careful BFGS updating) |
| Lecture 18 | Section 13.1 to 13.2 (Active constraints and LICQ) |
| Lecture 19 | Section 13.3 (Convex Problems) |
| Lecture 20 | Section 13.4 (Complementarity) to 14.1 (QPs via Active Set Method) |
| Lecture 21 | Section 14.2 (SQP) to 14.4 (Interior Point methods) |
| Lecture 22 | Section 14.4 (Barrier problem interpretation, SCP) to 15.5 (Simultaneous optimal control) |
| Lecture 23 | Problem reformulations and useful function approximations |
| Lecture 24 | Summary of the course |
Exercises
The exercises are based partially on pen and paper and partially on Python (see below).
Sheet (pdf) | Material (code) | Solution (pdf & code) |
| Exercise 1 - Introduction to CasADi, optimization problems | template1 | solutions1 |
| Exercise 2 - Convexity, Duality, and Fitting problems | None | - |
| Exercise 3 - coming soon | - | - |
| Exercise 4 - coming soon | - | - |
| Exercise 5 - coming soon | - | - |
If you find typos or mistakes in the exercises, please report them to Léo Simpson.
Further Material
- Books
- Jorge Nocedal and Stephen J. Wright, Numerical Optimization, Springer, 2006
- Stephen Boyd and Lieven Vandenberghe, Convex Optimization, Cambridge Univ. Press, 2004
- Amir Beck, Introduction to Nonlinear Optimization, MOS-SIAM Optimization, 2014.
- sample exam recent
Project
Here you can find the project guidelines.
Software
The exercises lean heavily on the open source tool CasADi, which offers an interface to Python, Matlab, and Octave. The computer exercise templates as well as the code solutions will be published in Python only.
Python is one of the major programming languages and via libraries such as NumPy and SciPy a common choice for scientific computing. Apart from CasADi, we will use the libraries NumPy, SciPy, and Matplotlib. If you are completely new to Python, you may want to check out the Anaconda distribution.
CasADi is a symbolic framework for algorithmic differentiation and numerical optimization. In order to install CasADi, follow the instructions here.
If you are using pip, you can simply pip install casadi in your preferred run' pip install casenvironment. If you are using Anaconda / conda for managing your environments, we suggest to first conda install pip inside a conda environment, followed by pip install casadi.