Optimization
Optimering

FMAN61F, 7.5 credits

Valid from: Autumn 2021
Decided by: Professor Thomas Johansson
Date of establishment: 2021-09-07

General Information

Division: Mathematics
Course type: Course given jointly for second and third cycle
The course is also given at second-cycle level with course codes: FMAN61, MATC61
Teaching language: English

Aim

In many applications of mathematics, e.g. image analysis, control theory and time series analysis, an essential step is to choose the parameters in a model so that it fits given data as well as possible. One wants to minimize the error, measured in some way, which may be considered as a function of several variables – the parameters – that may have to satisfy further conditions – constraints. The aim of the course is to make the doctoral student familiar with the most common methods for solving optimization problems in which the parameters may vary continuously.

Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• in his/her own words, be able to describe the optimization algorithms, for problems with and without constraints, encountered in the course, and their properties.
• be able to outline those the theory of convex sets and convex functions that is included in the course, and be able to state and derive the most important theorems on convexity.
• be able to give examples of how to to make use of convexity in the treatment of an optimization problem.
• be able to outline the Kuhn-Tucker Theory and be able to state and derive the most important theorems therein.

Competences and Skills

For a passing grade the doctoral student must

• be able to demonstrate an ability to solve optimization problems within the framework of the course.
• be able to demonstrate an ability to handle optimization problems using a computer.
• be able to demonstrate an ability to, in the context of problem solving, develop the theory somewhat further.
• be able to describe the connections between different concepts in the course, with proper terminology and in a well structured and logically consistent manner,.
• with proper terminology, suitable notation, in a well structured way and with clear logic be able to describe the solution to a mathematical problem and the theory within the framework of the course.

Course Contents

Quadratic forms and matrix factorisation. Convexity. Separatting planes and Farkas' lemma. The theory of optimization with and without constraints: Lagrange functions, Kuhn-Tucker theory. Duality. Methods for optimization without constraints: line search, steepest descent, Newton methods, conjugate directions, non-linear least squares optimization. The Nelder-Mead search method without derivatives. Methods for optimization with constraints: linear optimization, the simplex method, quadratic programming, penalty and barrier methods.

Course Literature

Böiers, L.: Mathematical Methods of Optimization. 2010. ISBN 9789144070759.

Instruction Details

Types of instruction: Lectures, seminars, laboratory exercises, exercises

Examination Details

Examination formats: Written exam, written report. Programming exercise with written report.
Grading scale: Failed, pass
Examiner:

Admission Details

Assumed prior knowledge: Calculus and linear algebra. Sufficient background is provided, e.g., by the courses FMAA05, FMA430, and FMAF05 or FMAF10.

Further Information

Replaces FMAN60F.

Course Occasion Information

Contact and Other Information

Course coordinators:
Web page: http://www.maths.lth.se/course/bigopt/