Hi all,
Gilles has started adding an API for simple bounds constraints in
optimization, which will soo be available for both Bobyqa and CMA-ES, as
both method naturally support such constraints.
I wondered if we could take the opportunity to set up a poor man
implementation in the abstract base class for optimizers that do not
support constraints, such as Nelder-Mead and Virginia Torczon's
multi-directional algorithm. Gilles work on the proper API plus this
poor man implementation would allow solving the old issue MATH-196 I
created 3 years and an half ago ...
A basic implementation can be based on an array of mappers inserted
between the user variables and the variables the optimizer manages. For
each component of the current point x[i] considered in user space, we
would map it to an optimizer variable y[i] using:
for (int i = 0; i < n; ++i) {
y[i] = mapper[i].encode(x[i]);
}
and a similar setting for retrieving x[i] from y[i] using a decode
function. The x[i] variables would be subject to bounds (some may be
bounded and others not bouded), whereas y[i] would all be unbounded.
Mapper would be an interface with four implementations, depending on bounds.
NoBoundsMapper would be used for components x[i] which are nturally
unbounded, encode and decode would be the identity function.
LowerBoundMapper would be used for components x[i] that must be larger
than a lower bound a:
encode(x) = ln(x - a)
decode(y) = a + exp(y).
UpperBoundMapper would be used for components x[i] that must be lesser
than an upper bound b:
encode(x) = -ln(b -x)
decode(y) = b - exp(-y).
LowerUpperBoundMapper would be used for components x[i] that must be
betwee lower bound a and upper bound b:
encode(x) = ln((x - a) / (b - x))
decode(y) = (a + b exp(y)) / (1 + exp(y))
Due to numerical considerations and the fact infinity can be
represented, I'm not sure if the bounds can be reached or not, it
probably depends on how we implement the encode/decode functions.
For sure, it does not replace proper handling of bounds, but it would be
quite useful. It is what we suggested as a fallback solution to several
users who needed constrained optimization.
If you agree with this, I could implement it in the next few days, so
Gilles could set up the proper handling for CMA-ES and fine-tune the API.
What do you think about this ?
Luc
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