# Oxford University MSc course on Numerical Optimization

 MSc in Mathematical Modelling & Scientific Computing MSc in Applied and Computational Mathematics Hilary Term 2005 Nick Gould - (n.i.m.gould@rl.ac.uk)

### 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] 3-4. Line-search methods for unconstrained minimization. [key points: 2, 4 slides per page] [notes: 1, 2 sides per page] 5-8. Trust-region methods for unconstrained minimization. [key points: 2, 4 slides per page] [notes: 1, 2 sides per page] 9-10. Interior-point methods for constrained minimization. [key points: 2, 4 slides per page] [notes: 1, 2 sides per page] 11-12. 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.

### Assignment

Those of you who are taking the course for "credit" need to write a shortish report on some optimization-related topic of interest to you. Some possiblities are given here, but I am quite happy if you would prefer to write about something else ... as long as you check with me first!

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) 1996

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

• 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: