Numerical Optimization (NO)
Course content
Numerical optimization 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 optimization like:
inverse kinematics in robotics, image segmentation and registration
in medial imaging, protein folding in computational biology, stock
portfolio optimization, motion planing and many more.
This course will build up a toolbox of numerical optimization
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 in science.
This course teaches the basic theory of numerical optimization
methods. The focus is on deep learning of how the methods covered
during the course works. Both on a theoretical level that goes into
deriving the math but also on an implementation level with focus on
computer science and good programming practice.
There will be weekly programming exercises where students will
implement the algorithms and methods introduced from theory and
apply their own implementations to casestudy problems like
computing the motion of a robot hand or fitting a model to highly
nonlinear data or similar problems.
The topics covered during the course are:
 First order optimality conditions, KarushKuhnTucker Conditions, Taylors Theorem, Mean Value Theorem.
 Nonlinear Equation Solving: Newtons Method, etc..
 Linear Search Methods: Newton Methods, QuasiNewton Methods, etc..
 Trust Region Methods: LevenbergMarquardt, Dog leg method, etc..
 Linear Least squares fitting, Regression Problems, Normal Equations, etc.
 And many more...
MSc programme in Computer Science
MSc Programme in Bioinformatics
Knowledge of:
 Theory of gradient descent method
 Theory of Newton and Quasi Newton Methods
 Theory of Thrust Region Methods
 Theory of quadratic programming problems
 First order optimality conditions (KKT conditions)
Skills to:
 Apply numerical optimization problems to solve unconstrained and constrainted minimization problems and nonlinear root search problems.
 Reformulate one problem type into another form such as root search to minimization and vice versa
 Implement and test numerical optimization methods
Competences to:
 Evaluate which numerical optimization methods are best suited for solving a given optimization problem
 Understand the implications of theoretical theorems and being able to analyze real problems on that basis
Mixture of study groups and project group work with handins.
See Absalon when the course is set up.
It is expected that students know how to install and use Matlab
by themselves. It is also 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.
 ECTS
 7,5 ECTS
 Type of assessment

Continuous assessmentThe exam is based on 57 written assignments and at least one oral presentation in class.
 Marking scale
 7point grading scale
 Censorship form
 No external censorship
Several internal examiners
Criteria for exam assessment
In order to achieve the highest grade 12, a student must be able to
 Derive Newton's method for nonlinear equations
 Derive Newton's method for constrained minimization problems
 Derive first order optimality conditions for a minimization problem
 Implement computer programs that can solve the selected problems presented during the course.
 Account for how the selected problems presented during the course is reformulated into mathematical models such as nonlinear equations or constrained minimization problems.
Single subject courses (day)
 Category
 Hours
 Exercises
 72
 Preparation
 50
 Project work
 84
 English
 206
Kursusinformation
 Language
 English
 Course number
 NDAA09009U
 ECTS
 7,5 ECTS
 Programme level
 Full Degree Master
 Duration

1 block
 Schedulegroup

C
 Capacity
 No limit
 Studyboard
 Study Board of Mathematics and Computer Science
 Department of Computer Science
Course responsible
 Sune Darkner (76b687972756c79476b7035727c356b72)
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