BA
in Mathematics, Section C
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Hilary Term 2006 Nick Gould |
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.
Who is teaching?
- Lecturer: Nick Gould, Comlab office 217, email
- Class Tutor: Denis Zuev, Maths Institute office DH03, email
How is this course organised?
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 weeks 3-8 of the term. Lectures are taking place in weeks 1-8 of HT05 on Mondays and Wednesdays from 2-3pm in the Fox Room in the Comlab. Classes also take place in the Fox Room on Mondays [10-12 TBC]. Solutions have to be put in Denis Zuev's pigeon-hole at the Maths Institute by 9AM on Monday directly before the class. See also the course outline slides.
Synopsis
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] 2-3. Optimality conditions and why they are important. [key poimts: 2, 4 slides per page] [notes: 1, 2 sides per page] 4-6. Line-search methods for unconstrained minimization. [key points: 2, 4 slides per page] [notes: 1, 2 sides per page] 7-9. Trust-region methods for unconstrained minimization. [key points: 2, 4 slides per page] [notes: 1, 2 sides per page] 10-11. Active-set methods for linearly-constrained minimization. [key points: 2, 4 slides per page] [notes: 1, 2 sides per page] 12. Penalty and augmented Lagrangian methods for constrained minimization. [key points: 2, 4 slides per page] [notes: 1, 2 sides per page] 13-14. Interior-point methods for constrained minimization. [key points: 2, 4 slides per page] [notes: 1, 2 sides per page] 15-16. 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 an accompanying paper.
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!
In addition, I will provide problem sheets and sample exams for the revision classes in TT06. All materials are posted in PDF format. In order to display it, you need the Acrobat software, which is installed on almost all PCs and workstations. If you are experiencing problems downloading some of the materials, please contact me via email.
Problem sheets
- Problem set 1: on the material of lectures 1-3.
- Problem set 2: on the material of lectures 4-6.
- Problem set 3: on the material of lectures 7-9.
- Problem set 4: on the material of lectures 10-12.
- Problem set 5: on the material of lectures 13-14.
- Problem set 6: on the material of lectures 15-16.
Revision material
Revision classes
- There will be two revision classes to discuss the sample exams 1 and 2. Details to follow.
Reading List
Last time this course ran, Raphael Hauser produced an excellent set of acompanying lecture notes.
The following books contain useful background material. The first is particularly recommended:
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 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 15 January 2006 at 09:54