# NDAA09009U Numerical Optimisation (NO)

Volume 2023/2024
Education

MSc Programme in Bioinformatics
MSc Programme in Computer Science
MSc Programme in Physics
MSc Programme in Statistics
MSc Programme in Mathematics-Economics

Content

Numerical optimisation is a useful computer tool in many disciplines like image processing, computer vision, machine learning, bioinformatics, eScience, scientific computing and computational physics, computer animation and many more. A wide range of problems can be solved using numerical optimisation like; inverse kinematics in robotics, image segmentation and registration in medical imaging, protein folding in computational biology, stock portfolio optimisation, motion planning and many more.

This course will build up a toolbox of numerical optimisation methods which the student can use when building solutions in his or her future studies. Therefore this course is an ideal supplement for students coming from many different fields of science.

This course teaches the basic theory of numerical optimisation methods. The focus is on deep understanding, and how the methods covered during the course works. Both on a theoretical level that goes into deriving the math but also on an implementational level focusing on computer science and good programming practice.

There will be weekly programming exercises where students will implement the algorithms and methods introduced from theory on their own case-study problems like computing the motion of a robot hand or fitting a model to highly non-linear data or similar problems.

The topics covered during the course are:

• First-order optimality conditions, Karush-Kuhn-Tucker Conditions, Taylors Theorem, Mean Value Theorem.
• Nonlinear Equation Solving: Newtons Method, etc.
• Linear Search Methods: Newton Methods, Quasi-Newton Methods, etc.
• Trust Region Methods: Levenberg-Marquardt, Dog leg method, etc.
• Linear Least-squares fitting, Regression Problems, Normal Equations, etc.
• And many more...
Learning Outcome

Knowledge of

• The theory of convex and non-convex optimisation
• The theory of Newton and Quasi-Newton Methods
• The theory of Trust Region Methods
• First-order optimality conditions (KKT conditions)

Skills in

• Applying numerical optimisation problems to solve unconstrained and constrained minimisation problems and nonlinear root search problems
• Reformulating one problem type into another form - for example reformulating constrained convex problems into unconstrained non-convex problems
• Implementing and testing numerical optimisation methods

Competences to

• Evaluate which numerical optimisation methods are best suited for solving a given optimisation problem
• Understand the implications of theoretical theorems and being able to analyse real problems on that basis

See Absalon when the course is set up.

The programming language used in the course is Python. It is expected that students know how to install and use Python, Numpy, Scipy and Matplotlib by themselves.

It is expected that students know what matrices and vectors are and that students are able to differentiate vector functions.

Theorems like fundamental theorem of calculus, mean value theorem or Taylor's theorem will be touched upon during the course. The inquisitive students may find more in depth knowledge from Chapters 2, 3, 5, 6 and 13 of R. A. Adams, Calculus, 3rd ed. Addison Wesley.

Academic qualifications equivalent to a BSc degree is recommended.
Mixture of study groups and project group work with hand-ins and small lectures.
The focus is on flipped-classroom teaching.
• Category
• Hours
• Lectures
• 10
• Preparation
• 40
• Exercises
• 72
• Project work
• 84
• Total
• 206
Written
Oral
Individual
Continuous feedback during the course of the semester
Credit
7,5 ECTS
Type of assessment
Continuous assessment
Type of assessment details
The assessment is based on 5-7 written group assignments (with individual contributions noted). All students must hand in all assignments individually so that the assignments can be individually approved. The final grade is an unweighted average of the five best assignments.
Aid
All aids allowed
Marking scale
7-point grading scale
Censorship form
No external censorship
Several internal examiners
Re-exam

The re-exam is a resubmission of the written assignments and a 15-minute oral examination without preparation.

The assignments must be submitted no later than two weeks before the re-exam date i.e. the oral examination.

##### Criteria for exam assesment

See Learning Outcome.