[Rd] Optimization in R

[Andrew Clausen]

Hi all,

I've been working on improving R's optim() command, which does general purpose
unconstrained optimization.  Obviously, this is important for many statistics
computations, such as maximum likelihood, method of moments, etc.  I have
focused my efforts of the BFGS method, mainly because it best matches my
current projects.

Here's a quick summary of what I've done:
 * implemented my own version of BFGS in R,
	http://www.econ.upenn.edu/~clausen/computing/bfgs.zip
 * written a wrapper for the GNU Scientific Library's optimization function,
multimin(),
	http://www.econ.upenn.edu/~clausen/computing/multimin.zip
 * written some tricky functions to compare implementations,
	http://www.econ.upenn.edu/~clausen/computing/tests.zip

My own implementation has several advantages over optim()'s implementation
(which you can see in the vmmin() function in
	https://svn.r-project.org/R/trunk/src/main/optim.c)
 * the linesearch algorithm (More-Thuente) quickly finds a region of interest
to zoom into.  Moreover, it strikes a much better balance between finding
a point that adequately improves upon the old point, but doesn't waste too
much time finding a much better point.  (Unlike optim(), it uses the standard
Wolfe conditions with weak parameters.)
 * the linesearch algorithm uses interpolation, so it finds an acceptable
point more quickly.
 * implements "box" constraints.
 * easier to understand and modify the code, partly because it's written in R.

Of course, this comes at the (slight?) overhead cost of being written in R.

The test suite above takes the first few functions from the paper

        Moré, Garbow, and Hillstrom, "Testing Unconstrained
        Optimization Software", ACM Trans Math Softw 7:1 (March 1981)

The test results appear below, where "*" means "computed the right solution",
and "!" means "got stuck".

test                    optim           clausen         gsl
--------------------------------------------------------------
bard                                                    !
beale
brown-scaled
freudenstein-roth
gaussian                                                *
helical-valley          *               *
jennrich-sampson                                        *
meyer                                                   *
powell-scaled                           *
rosenbrock                              *

The table indiciates that all three implementations of BFGS failed to compute
the right answer in most cases.  I suppose this means they are all quite
deficient.  Of course, this doesn't imply that they perform badly on real
statistics problems -- but in my limited experience with my crude econometric
models, they do perform badly.   Indeed, that's why I started investigating in
the first place.

For what it's worth, I think:
 * the optimization algorithms should be written in R -- the overhead is
small compared to the cost of evaluating likelihood functions anyway, and is
easily made up by the better algorithms that are possible.
 * it would be useful to keep a repository of interesting optimization
problems relevant to R users.  Then R developers can evaluate "improvements".

Cheers,
Andrew