Introduction

Optimization deals with the problem of minimising or maximising a mathematical model of an objective function such as cost, fuel consumption etc. under a set of side constraints on the domain of definition of this function. Optimization theory is the study of the mathematical properties of optimization problems and the analysis of algorithms for their solution. The aim of this course is to provide an introduction to nonlinear continuous optimization specifically tailored to the background of mathematics students.

The major pre-requisites for the course will be some knowledge of both linear algebra and real analysis, while an appreciation of methods for the numerical solution of linear systems of equations will be helpful.

Intended Synopsis

• Lecture 1: A gentle introduction.

• Lectures 2-3: Optimality conditions and why they are important.

• Lectures 4-6: Line-search methods for unconstrained minimization.

• Lecture 7-9: Trust-region methods for unconstrained minimization.

• Lectures 10-11: Active-set methods for linearly-constrained minimization.

• Lecture 12: Penalty and augmented Langrangian methods for constrained minimization.

• Lectures 13-14: Interior-point methods for constrained minimization.

• Lectures 15-16: Sequential quadratic programming (SQP) methods for constrained minimization.

Reading List

Notes will be provided. the following books contain useful background material.

• J. Nocedal and S. Wright, Numerical Optimization, Springer Verlag 1999

• J. Dennis and R Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, (republished by) SIAM (Classics in applied Mathematics 16) 1996

• P. Gill, W. Murray and M. Wright, Practical Optimization, Academic Press, 1981

• R. Fletcher, Practical Methods of Optimization, 2nd edition Wiley, 1987, (republished in paperback in 2000)

• A. Conn, N. Gould and Ph. Toint, Trust-Region Methods, SIAM, 2000

Course Organization

The course consists of 16 lectures. There are problem sheets for students studying for Part C, and classes will be run to discuss these. MSc students will be expected to write a project related to the course.