Numerical Optimization

Lectures: Prof. Dr. Moritz Diehl,          Exercises: Florian Messerer 

The course’s aim is to give an introduction into numerical methods for the solution of 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 inverted classroom and based on four pillars:

  • a course manuscript
  • lecture recordings, and
  • exercises, that are accompanied with solutions and solution recordings, and
  • weekly alternating Q&A and exercise sessions to discuss the course contents

Contact: moritz.diehl@imtek.uni-freiburg.de, florian.messerer@imtek.uni-freiburg.de

*** This page is intended for the course as taught in the winter semester 2021/22 at the University of Freiburg. For a timeless version with a focus on self study of the material see here. ***

Organization of the course

The course is organized as 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 do a project to get an additional 3 ECTS, i.e., a total of 9 ECTS for course+project. For more information please contact Florian Messerer.

Hybrid meetings: We will meet every Friday, 10 to 12. We will use a hybrid format, i.e., it will be possible to attend the meetings both in presence (3G rules apply) and remotely via Zoom:

These meetings are alternatingly dedicated to either Q&A sessions with Prof. Diehl or exercise sessions with the teaching assistant (see below). and will not be recorded.

Ilias: There is also an Ilias course, though most material will be published on the page you are currently viewing. In Ilias, we provide a forum for discussion of any questions you have related to the course, be it organization, content or exercises. Please feel free to open new topics and to answer questions of your fellow students. Further, the mid term quiz will be published on Ilias (see below).
Join Ilias course

Lecture recordings: The lecture recordings were already created in a past semester. There are 24 lectures of approximately 90 minutes each. 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, which you may find in the materials section below. We will provide printed versions.

Exercises: The exercises are mainly computer based. Computers with MATLAB and CasADi installed are required to solve them (see below for details). The exercises are voluntary (though of course we strongly recommend to solve them). Nonetheless 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 especially, you can state so on your solution sheet. To hand them in, send them in an email to florian.messerer@imtek.de.

Q&A sessions: Every second week there will be a virtual 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 not be used to show the solutions, but to discuss 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: Some time before christmas, we will publish a quiz on Ilias, with questions covering the course contents so far. It is obligatory that you pass this quiz until 10.12., 23:59, but 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 sheets (i.e., 4 pages) of self-chosen content are allowed (handwritten). For students from M.Sc. MSE / ESE and B.Sc. Math, this exam is graded. Students from the M.Sc. Math need to pass the written exam in order to take the graded 11ECTS oral exam. Unmentioned special cases: Anyone 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&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  Prepare  Deadlines  Check in
 22.10.  Intro  up to including Chap. 2.7  Lec. 1, 2  one question    link
 29.10.  Ex  Ex 1  Lec. 3, 4  one question    link
 05.11.  Q&A  up to including Chap. 6.5  Lec. 5, 6  one question  Ex 1 (voluntary)  link
 12.11.  Ex  Ex 2; sol Ex 1  Lec. 7, 8  one question    link
 19.11.  Q&A  up to including Chap. 9.3  Lec. 9, 10  one question  Ex 2 (voluntary)  link
 26.11.  Ex  Ex 3; sol Ex 2  Lec. 11, 12  one question    link
 03.12.  Q&A  up to including Chap. 11.2  Lec. 13, 14  one question  Ex 3 (voluntary)  link
 10.12.  Ex  Ex 4; sol Ex 3  Lec. 15, 16  one question  mid term quiz (obligatory, until 23:59)  link
 17.12.  Q&A  up to including Chap. 13.2  Lec. 17, 18  one question  Ex 4 (voluntary)  link
 24.12. *** *** Christmas Break *** ***   ***  ***
 31.12.
 07.01.
 14.01.  Ex  Ex 5; sol Ex 4  Lec. 19, 20  one question    link
 21.01.  Q&A  up to including Chap. 15.5  Lec. 21, 22  one question  Ex 5 (voluntary)  link
 28.01.  Ex  projects; sol Ex 5  Lec. 23, 24  one question    link
 04.02.  Q&A  All course content    one question    link
 11.02.  pres        project presentations  link
 25.02.  none  no session, just a deadline      project reports  -

 

 

Manuscript

The lectures are closely following a course manuscript draft.

Numerical optimization (draft manuscript) by M. Diehl

Lectures

The video recordings correspond each to approximately 90 minutes, 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 Matlab + CasADi (see below).

 Sheet (pdf)  Material (code) Solution (video)  Solution (material)
 Exercise 1 - Introduction to CasADi, Convex Optimization      
 Exercise 2 - Duality and Fitting Problems      
 Exercise 3 - Unconstrained Newton-type Optimization, Globalization      
 Exercise 4 - Calculation of Derivatives, Equality Constrained Optimization      
 Exercise 5 - Inequality Constrained Optimization      

 

Further Material

 

Project

Details on the optional project will be added here.

 

Matlab and CasADi installation

MATLAB is an environment for numerical computing based on a proprietary language that allows one to easily manipulate matrices and visualize data which will be very helpful in prototyping the algorithms presented during the lectures of this course. The University of Freiburg offers a free-of-cost license to students and staff which can be obtained following the instructions here. In order to be able to complete the exercises of this course, you will need a working installation of MATLAB. Follow the instructions at the provided link in order to install the software package.
 
CasADi is a symbolic framework for algorithmic differentiation and numerical optimization. In order to install CasADi, follow the instructions here. Download the binaries for your platform and, after having extracted them, add their location to MATLAB's path. To test your installation run the simple example described at the provided link. If successful, save the path by executing the command savepath. In this way, the location of the binaries will be known even after restarting MATLAB.