(Created 2009-08-11.)
 OPTIMIZATION FMA051

Higher education credits: 6. Grading scale: TH. Level: A (Second level). Language of instruction: The course will be given in English on demand. Alternative for: I3. Optional for: D4, E4, E4pe, E4ra, F4, F4sfm, F4tmb, F4tvb, Pi3, Pi3bm, Pi3bs, Pi3fm, Pi3mrk, Pi3sbs. Course coordinator: Director of Studies, Lars-Christer Böiers, Lars_Christer.Boiers@math.lth.se, Matematik. Recommended prerequisits: Basic university studies in calculus and linear algebra. Assessment: Written test comprising theory and problems. Two computer exercises and one project. Home page: http://www.maths.lth.se/matematiklth/vitahyllan/vitahyllan.html.

Aim
The aim of the course is to present basic optimisation theory, and to give an overview of the most important methods and their practical use.

Knowledge and understanding
For a passing grade the student must

be familiar with and in his/her own words be able to describe the optimisation algorithms encountered in the course, with and without constraints, and their properties.

be familiar with the theory of convex sets and convex functions, and be able to state and derive the most important theorems on convexity.

be familiar with Kuhn-Tucker Theory and be able to state and derive the most important theorems therein.

Skills and abilities
For a passing grade the student must

be able to show capability to solve optimisation problems within the framework of the course.

be able to show capability to handle optimisation problems using a computer.

be able to show capability to, in the context of problem solving, develop the theory somewhat further.

with proper terminology, well structured and with clear logic, be able to describe the connections between different concepts in the course.

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.

Contents
Quadratic forms and matrix factorisation. Convexity. The theory of optimisation with and without constraints: Lagrange functions, Kuhn-Tucker theory. Duality. Methods of optimisation without constraints: line search, steepest descent, Newton methods, conjugate directions, non-linear least squares optimisation. Methods of optimisation with constraints: linear optimisation, the simplex method, quadratic programming, penalty and barrier methods.

Literature
Böiers, L-C: Lectures on Optimisation. KF-Sigma 2004.
Department of Mathematics: Exercises in Optimisation. KF-Sigma 2004.
Department of Mathematics: Computer Laboratory Exercises in Optimisation.