## Introduction

The optimization (minimization or maximization) of a function of a number of unknown parameters (possibly) subject to constraints is, along with the solution of differential equations and linear systems, one of the three corner-stones in computational applied mathematics. In this course, I shall aim to introduce the central ideas behind algorithms for the numerical solution of differentiable optimization problems. I intend to present key methods for both unconstrained and constrained optimization, as well as providing theoretical justification as to why they succeed.

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.

## Synopsis

Brief content of the lectures - PDF copies of the key points for each lecture, along with notes giving details of proofs of results, will be made available in advance:

1. A gentle introduction. [key poimts: 2, 4 slides per page] 2. Optimality conditions and why they are important. [key poimts: 2, 4 slides per page] [notes: 1, 2 sides per page] 3. Review of line-search methods for unconstrained minimization. [key points: 2, 4 slides per page] [notes: 1, 2 sides per page] 4-6. Trust-region methods for unconstrained minimization. [key points: 2, 4 slides per page] [notes: 1, 2 sides per page] 7-8. Interior-point methods for constrained minimization. [key points: 2, 4 slides per page] [notes: 1, 2 sides per page] 9-10. Sequential quadratic programming (SQP) methods for constrained minimization. [key points: 2, 4 slides per page] [notes: 1, 2 sides per page] Full lecture notes are available as a paper.

## Assignments

There are no formal assignments, but at the end of each section, you might wish to attempt some of the "tutorial" questions. Model solutions are available for you to check your answers.

## Previous examination questions

Here is a set of previous exam questions and model solutions:

specimen (solution) 2003 (solution) 2004 (solution) ## Reading List

Students are encouraged to read at least one of the seminal papers in the field. In addition, the following books contain useful background material:

J. Dennis and R. Schnabel,

Numerical Methods for Unconstrained Optimization and Nonlinear Equations,(republished by) SIAM (Classics in Applied Mathematics 16) 1996J. Nocedal and S. Wright,

Numerical Optimization,Springer Verlag 1999P. Gill, W. Murray and M. Wright,

Practical Optimization,Academic Press 1981R. Fletcher,

Practical Methods of Optimization,2nd edition Wiley 1987, (republished in paperback 2000)A. Conn, N. Gould and Ph. Toint,

Trust-Region Methods,SIAM 2000## WWW sites

Optimization, at its best, is featured on the following WWW sites:

- Linear programming frequently-asked questions
- Nonlinear programming frequently-asked questions
- The Optimization Technology Center with its guide to optimization software
- The Network-Enabled Optimization System(NEOS) - solve optimization problems online
- A Decision Tree for Optimization Software, and associated Benchmarks for Optimization Software
- Optimization Online
- INFORMS resouces
- Mathematical Programming Society
- SIAM
## Last updated 4 January 2006 at 08:25 GMT