[R] Dealing with -Inf in a maximisation problem.

[Paul Gilbert]

>
> I am trying to deal with a maximisation problem in which it is possible
> for the objective function to (quite legitimately) return the value
> -Inf,

(Just to add to the pedantic part of the discuss by those of us that do 
not qualify as younger and wiser:)

Setting log(0) to -Inf is often convenient but really I think the log 
function is undefined at zero, so I would not refer to this as "legitimate".

>which causes the numerical optimisers that I have tried to fall over.

In theory as well as practice. You need to have a function that is 
defined on the whole domain.

>
> The -Inf values arise from expressions of the form "a * log(b)", with b
> = 0.  Under the *starting* values of the parameters, a must equal equal
> 0 whenever b = 0, so we can legitimately say that a * log(b) = 0 in

This also is undefined and not "legitimate". I think there is no reason 
it should be equal zero. We tend to want to set it to the value we think 
of as the "limit": for a=0 the limit as b goes to zero would be zero, 
but the limit of a*(-inf) is -inf as a goes to zero.

So, you really do need to avoid zero because your function is not 
defined there, or find a redefinition that works properly at zero. I 
think you have a solution from another post.

Paul

> these circumstances.  However as the maximisation algorithm searches
> over parameters it is possible for b to take the value 0 for values of
> a that are strictly positive.  (The values of "a" do not change during
> this search, although they *do* change between "successive searches".)
>
> Clearly if one is *maximising* the objective then -Inf is not a value of
> particular interest, and we should be able to "move away".  But the
> optimising function just stops.
>
> It is also clear that "moving away" is not a simple task; you can't
> estimate a gradient or Hessian at a point where the function value is -Inf.
>
> Can anyone suggest a way out of this dilemma, perhaps an optimiser that
> is equipped to cope with -Inf values in some sneaky way?
>
> Various ad hoc kludges spring to mind, but they all seem to be fraught
> with peril.
>
> I have tried changing the value returned by the objective function from
> "v" to exp(v) --- which maps -Inf to 0, which is nice and finite.
> However this seemed to flatten out the objective surface too much, and
> the search stalled at the 0 value, which is the antithesis of optimal.
>
> The problem arises in a context of applying the EM algorithm where the
> M-step cannot be carried out explicitly, whence numerical optimisation.
> I can give more detail if anyone thinks that it could be relevant.
>
> I would appreciate advice from younger and wiser heads! :-)
>
> cheers,
>
> Rolf Turner
>
> -- Technical Editor ANZJS Department of Statistics University of
> Auckland Phone: +64-9-373-7599 ext. 88276