# 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.Gould@rl.ac.uk)

### 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!

You might choose any one of these topics:

Using the seminal papers as a starting point,

1. consider how relevant quasi-Newton methods are today, or

2. investigate non-monotone linesearch and trust region methods, or

3. investigate methods that are likely to be appropriate if the number of unknowns is huge.

Alternatively,

4. Investigate methods that are appropriate when the objective function is the sum of squares of nonlinear functions, and that exploit this particular structure.

5. How might the methods we have been considering generalize to infinite dimensions?

6. have a look at the Minimum Surface Problem (number 17, p39) in Dolan and Moré's paper on optimization problems. Consider a variation of this problem, where the obstacle is required to be at least 1.0 within a circle of radius 0.25, and larger than zero outside the circle.

What does the solution look like when the circle in centred at (0.5,0.5)? Now consider the solution as the centre of the circle is allowed to move anywhere within the unit square (and for which any any obstacle points outside the square are ignored). What is the largest possible area of the surface?

I will be looking to see how you formulate and solve the problem, and which method/algorithm(s) you use; in particular, please describe the method/theory behind any software used.

7. If you find any of the other of Dolan and Moré's problems to be of interest to you, you might try variations on one of these themes.

8. If you have a "pet" project you would like to investigate, please discuss it with me, and we will see if it is suitable.