Hi Grey, and others,
I wish to consider a whole family of optimization problems which are
defined as follows:
the objective function f: [0,1]^n -> R
the n parameters are partitioned in m sets so that each set of
parameters represents a distribution of probability.
So, for a very small example, let us suppose that n = 3, m = 1, and
parameters are denoted a, b, and c. Then, we have the constraint that
a+b+c=1.
Performing a gradient algorithm, we need the gradient of f wrt each
parameter. However, the gradients are not independent: to keep this
a+b+c=1 relationship, df/da is not independent of df/db and df/dc.
Simply updating one parameter (say a) using its gradient (df/da) is not
correct: the space of parameters is not Euclidian because the variation
of one parameter involves some variation on the other 2, to keep this
a+b+c=1 constraint valid.
So my question is whether the algorithms available in nlopt take care of
this. I doubt it but I'd like to be sure.
Then, the next question is: how to take care of this relationship?
Thanks a lot,
Philippe
On 25/01/2017 18:52, Grey Gordon wrote:
Hi Philippe,
Is your problem to min_{a,b,c} f(a,b,c) s.t. a+b+c=1 for f:R^3 -> R? Do you
mean your function is non-Euclidean because it is mapping to some space other than
R?
Perhaps more concretely explaining your problem would help.
Best,
Grey
On Jan 25, 2017, at 11:18 AM, philippe preux<[email protected]> wrote:
Hi,
I am optimizing a differentiable function defined over a probability
distribution. That is, say the function to optimize has 3 parameters a, b and c
each being a probability and such that a + b + c = 1.
We know that optimizing each parameter independently from the other 2 is not
the best way to go as we do not take the a+b+c=1 constraint into consideration.
The solution is not to add this constraint to the problem via an equality
constraint; the issue is that the space is not Euclidian and that whenever one
computes the gradient wrt to a parameter (say a), the 2 others should also be
considered, to take the shape of the manifold on which I optimize into
consideration. It seems to me that directional derivatives, or natural
gradients are needed here.
So my question is: how to deal with such non Euclidian spaces with nlopt?
Thanks for any help,
Philippe
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