[R-SIG-Finance] DEoptim and guarantees

[alexios ghalanos]

On 24/08/2014 19:24, Enrico Schumann wrote:
> On Fri, 22 Aug 2014, alexios ghalanos <alexios at 4dscape.com> writes:
> 
>> 1. DEoptim is a nonlinear global optimization solver. Global
>> optimization is usually reserved for hard to solve non-convex
>> problems with many local minima. There is no guarantee of
>> optimality not even for convex problems, nor any idea of
>> whether the answer you are getting is anything other than a
>> local optimum.
> 
> There is no *mathematical* guarantee.  But that does not imply
> that you cannot use Differential Evolution (the method that
> DEoptim implements) with confidence.  Just because you cannot
> prove something does not mean that it is not the case.
> 
Accepted, but I'm sure you are not saying that a convex problem which
can be confidently and quickly solved by convex solvers should instead
be solved by differential evolution or other global optimization solvers?

> You do not need mathematical proofs to make meaningful statements
> about whether or how well an optimisation method works.[*] For a
> given model class (such as particular portfolio-selection
> models), you can run experiments.  Experimental results are no
> general proof, of course; but they are evidence of how a method
> performs for that particular type of model, and typically that is
> all that we care about when we apply a method.  In other words,
> you may not be able to mathematically prove that a method works,
> but you can have empirical evidence that is does.
Yes, but if the problem is convex, then there is one solution, and this
can usually be attained quite quickly with specialized convex solvers.
> 
> In practical optimisation, it is not useful to think of "the
> [optimal] solution" to a model, and "all the rest".  An
> appropriate way to think of it is "no solution, some solution, a
> better solution, an even better solution, ..."  and so on.  That
> is, think of "iterative improvement", not of optimisation.
> 
If by "practical optimization" you mean problems which are non-convex or
particularly difficult to solve (e.g. mixed integer), then perhaps. But
for convex problems, there is only one solution (by definition). Whether
that solution turns out to be sub-optimal in the out of sample, that is
down to uncertainty, quality of inputs etc.
However, I have seen a tendency to equate "practical" optimization, with
a simply lazy consideration of an optimization problem without making
the effort to see whether that problem can be put in a convex form.
Plug-and-play, without making some effort to understand the problem is
never, IMHO, a good way to do things.

> 
> [*] If you need an example other than Differential Evolution for
>     that, then look at Nelder--Mead.  You cannot prove anything,
>     and yet the method "just works".
>
Regards,

Alexios