(Created 2011-09-01.)

OPTIMIZATION | FMA051 |

**Aim**

The aim of the course is to present basic optimization 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 the theory of convex sets and convex functions, and be able to state and derive the most important theorems on convexity.

be aware of how to make use of convexity in the treatment of an optimization problem.

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 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.

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 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. Methods for optimization with constraints: linear optimization, the simplex method, quadratic programming, penalty and barrier methods.

**Literature**

Böiers, L-C: Lectures on Optimization. Studentlitteratur 2010.

Department of Mathematics: Computer Laboratory Exercises in Optimization.