- Scientific Computing Department, STFC-Rutherford Appleton Laboratory
- Chilton, Oxfordshire OX11 0QX, England
- nick.gould@stfc.ac.uk
- Telephone +44 (0)1235 445801

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 mathematicsl 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.

The course consists of lectures and classes, but no practicals. Students hand in solutions to six problem sheets which will be discussed in classes during the term.

Material for a example iteration of this course for scientists at RAL is available.

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: 1, 2, 4 slides per page] |

2-3. | Optimality conditions and why they are important. [key poimts: 1, 2, 4 slides per page] [notes: 1, 2 sides per page] |

4-6. | Line-search methods for unconstrained minimization. [key points: 1, 2, 4 slides per page] [notes: 1, 2 sides per page] |

7-9. | Trust-region methods for unconstrained minimization. [key points: 1, 2, 4 slides per page] [notes: 1, 2 sides per page] |

10-11. | Active-set methods for linearly-constrained minimization. [key points: 1, 2, 4 slides per page] [notes: 1, 2 sides per page] |

12. | Penalty and augmented Lagrangian methods for constrained minimization. [key points: 1, 2, 4 slides per page] [notes: 1, 2 sides per page] |

13-14. | Interior-point methods for constrained minimization. [key points: 1, 2, 4 slides per page] [notes: 1, 2 sides per page] |

15-16. | Sequential quadratic programming (SQP) methods for constrained minimization. [key points: 1, 2, 4 slides per page] [notes: 1, 2 sides per page] |

Full lecture notes are available as an accompanying booklet.

These notes form a self-contained introduction to the subject,
and constitute *compulsory reading assignments*.
Suggestions on how to improve the lecture notes are always very welcome!

The following books contain useful background material. The first is particularly recommended:

J. Nocedal and S. Wright,

*Numerical Optimization,*Springer Verlag 1999J. Dennis and R. Schnabel,

*Numerical Methods for Unconstrained Optimization and Nonlinear Equations,*(republished by) SIAM (Classics in Applied Mathematics 16) 1996P. 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

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 Optimization Society
- SIAM