For my diploma thesis I needed an easy-to-use optimisation algorithm which could minimise a given function. I had access to Matlab, but surprisingly, none of the supplied optimisation functions seemed to satisfy my needs: I wanted to globally minimise a non-linear function within boundaries without using gradients.

So I did a little research and found a Globalized Nelder-Mead method, which is a tuned version of the good old downhill simplex algorithm. Then I implemented it as a Matlab function. By the way: the function also works fine with Octave.

What does it do?

The Nelder-Mead algorithm, sometimes also called downhill simplex method, was originally published in 1965. It is an iterative algorithm for local, unconstrained minimisation of a non-linear function . In contrast to most other iterative algorithms, it does not rely on the derivative of the target function but only evaluates the function itself. This makes it extremely well suited for cases when the target function is not an algebraic term, but a simulation model which cannot be derived directly. It also does not approximate the function’s gradient but is based on geometrical projections of an n+1-point polygon—the simplex—in the n-dimensional parameter space of the target function. A simplex in two dimensions is a triangle, one in three dimensions is a tetrahedron. In general, the simplex can be represented as a matrix of its points.

The basic idea behind the simplex algorithm is that the worst point of is replaced by a better one with a smaller function value and to repeat this operation iteratively until a local optimum has been found. To achieve this, three different transformations are applied to the simplex which let it move in directions of descent and shrink around a local minimum. These transformations are called reflection, expansion and contraction.

In each iteration, first the worst point is reflected at the centroid of the remaining points. Depending on the quality (good or bad) it then replaces the worst point in the simplex. If the reflected point is not better, it is withdrawn and the other fallback operations are executed.

The globalized Nelder-Mead method just generalizes this idea to find a global minimum. Instead of initialising the simplex once, multiple simplices are initialised and each one finds one local minimum. There are some tweaks to make the location of these simplices better than random, but principally they are spread randomly within the allowed boundaries.

How do I use it?

Just copy the file gbnm.m from the download below into a directory where Matlab or Octave can find it. Then just call the function gbnm. A typical usage might look like this:

% Define test function and optimisation boundaries
f = @(x) norm(x).^2 - cos(norm(x));
xmin = [-10;-10];
xmax = -xmin;

xoptim = gbnm(f,xmin,xmax)
% Expected output: xoptim = [0.00.. ; 0.00..]

The function does not only return the found minimum and has an optional argument for tweaking the algorithm parameters:

[x, fval, output, options] = gbnm(fun,xmin,xmax[, options])

Refer to the output of help gbnm and the research paper for further detail. If something remains unclear or you have found a bug, feel free to comment!

Download

• gbnm.zip (v0.1 [2010-07-19] initial release)

But there is another, almost perfectly hidden feature which allows to typeset easy formulas as real text which respects text formatting styles: Equation fields.

Equation fields

Hit Ctrl+F9 and add the following construct: { EQ \f(a;b) }. Then place the cursor on or before this field and hit Shift+F9. You should see a fraction like this: .

The advantage of this technique over Word’s built-in formula editor is that these formulas are styled according to their surround text automatically. The omits much of the hassle involved in placement and handling of the formula objects.

There is just one thing you should not do: Don’t double click on them because they are converted to formula objects this way. If this happens by accident, use Undo (Ctrl+Z) to repair the damage. To edit an equation field, place the cursor on it and hit Shift+F9 again to unveil its code.

Commands

An equation field can contain any sequence of the following commands, mixed with plain text. Nesting is supported. Each command I find useful is presented with a short example of typical use.

	Description	Example
\a	Matrix. Available options: con determines number of columns, hs and vs horizontal and vertical spacing (in pt), the command \al, \ac and \ar allow to determine the horizontal alignment within the cells (\al is default). Example code: \a \co2 \hs3 (a;b;c;d)
\b	Braces. Places automatically sized braces around big expressions like fractions or roots. Example code: \b(\f(•;•))
\d	Positioning. Creates spaces and indentation. I don’t use it.
\f	Fraction. Example code: \f(a;b)
\i	Integral. Example code: \i(a;b;f(x)dx)
\l	List. I don’t use it.
\o	Overwrite. Places text in the second argument over the text in the first argument. Options \al, \ac and \ar determine their alignment. Example code: \o\ac(n;\s\up4(•))
\r	Root. Example code: \r(n;a)
\s	Super- and subscript. Available options are \don and upn. The first one creates subscripts, the second one superscripts. Example code: a\s\up4(n) b\s\do4(i)
\x	Frame. Places a framed box around its argument. Example code: \x(a)

Golden path

If you have to write a longer document, like a thesis, which contains vast amounts of formulas, get LaTeX! (Technical remark: the examples above are rendered using a server-based LaTeX-Renderer.)

Deskmodding is the art or the act of customizing the appearance of the graphical user interface. Usually this is done on the level of the operating system, but there are also single applications that allow to modify their appearance by so-called skins. For the operating system Windows it is not possible to easily change the default appearance as the selection of themes that can be selected is limited to a handful of designs like the classic Windows interface, Luna for Windows XP and Aero since Vista. Windows Media Center Edition brought a revamped version of Luna which became very popular.

But apart from those designs, Windows does not permit to select user-made themes. This is due to a check in the file uxtheme.dll in the system32 folder. There are dozens of downloads of patches that promise to replace the file with a patched version that permits so select whatever style one prefers. But it is easy to download the wrong patch or even get a virus on one’s computer this way. But there is an alternative: UxStyle.

UxStyle is an application that installs a small service under Windows that deactivates blocking of custom made themes on-the-fly. No changing of uxtheme.dll is necessary. Final information: A great repository for themes is the section Visual Styles on deviantART. Downloaded themes have to be placed in the folder Resources\Themes in the Windows installation folder.

One disadvantage of my how-to is that the data which is synchronised is transmitted without encryption. However, it is possible to somehow use SSL encryption for the data transfer, but so far I am afraid of the effort to reconfigure all again now that it works. Another disadvantage concerns the fact that currently only one calendar and one address book can be synchronised. While this restriction does not bother me now, it might render this solution useless for you.

1. Setting up the server

The first part of this how-to is setting up the funambol server. First download the latest version of the installer, then follow the instructions of the setup process. After the installation, the server should have been started (visible by the tray icon). If not, start it manually using the links in the start menu. If you want the server to be started automatically when you launch the computer after each reboot, copy the link into the “Startup” folder of your start menu.

Then launch the Administration Tool, click onFile > Login and enter the user “admin” with the default password “sa”. The first thing to do is to change it: Select the $Hostname > Users in the tree on the left. Then click the Search button on the right side to display all users in the list below. Double-click on the entry of the user “admin” and enter a new password (two times). Save.

To make your server reachable from other machines it is necessary to setup the URI from which it will be reached. You need to setup a port forwarding on port 8080 from your router to the local IP address of your server computer. In the Administration tool, you then have to enter Server settings the string “http://%your_ip%:8080/funambol/ds” while replacing %your_ip% with your external IP address or a domain provided by a service like DynDNS.

One last addition has to be done before Thunderbird calendars can be synched: Under Modules > foundation > FunambolFoundationConnector > PIM Calendar SyncSource you have to add a new SyncSource. Double click the entry PIM Calendar SyncSource on the left, then enter “ical” for both the source URI and the name and select “iCalendar” as the default type. All settings are also shown in this image:

Then click “Add” and check that your entry got also added to the tree on the left. Now your server should be ready for its first synchronisation. You can close the Administration Tool now.

2. Setting up a Thunderbird client

Just download the Funambol Mozilla Sync Client and install it as an add-on from within Thunderbird. The add-on can be accessed in the menu Tools > Funambol sync. Open it and select Tools > Options from the menu:

In the Account pane, enter exactly the URI you entered in your server before. For the username and password you can choose one arbitrarily for the first client you setup. For each new client you want to synchronise with you have to use the same username and password, of course.

In the pane Sync you have to select the address book and the calendar to be synchronised with the server and how often you want the synchronisation to be performed automatically. I chose 15 minutes and deactivated the synchronisation on startup. During setup (read: hunt for wrong settings) I stuck to manual synchronisation.

Close the configuration window and bravely click on Synch all. On the first synch, simply all your client’s data is transmitted to the server. On subsequent runs only changes will be transmitted.

On the second PC you just have to repeat the add-on installation, provide the same username and the synchronisation works.

3. Setting up the Windows Mobile client

Get the client *.cab file from the download page and transmit it to your phone. Execute it there to install it. The configuration is analogous to the Thunderbird client. I chose manual synchronisation for my mobile phone as I only run it when I have wireless LAN access. I only activated synchronisation of contacts, calendar and tasks — e-mail is taken care off by the IMAP servers.

Remarks

At first Thunderbird froze during each synchronisation. The cause for me was that I had used an *.cal-file for my private calendar. For the synchronisation to work I had to convert it to a calendar that is stored in Thunderbird’s profile folder (i.e. one with “moz-profile-calendar://?id=X” in the address field).

Update: there is one issue with my setup concerning the synchronisation between both Thunderbird clients: Reminders of events keep popping up on one client after having been confirmed on the other and vice versa. The problem is known to the developers and discussed in the developers’ board. As far as I understand the subject, a clean solution would require changes to the server itself which would take a long time to take effect. A “dirty fix” in the Thunderbird client could be possible, but so far no one with enough knowledge to do it wrote one. I keep you updated here if a fix is published.