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