Sampling for n-choose-k and linear equality

[bertiqwerty]

Currently, it is not possible to randomly sample from problems that have n-choose-k-constraints and linear equality constraints on the same variables.

[DavidWalz]

This is a hard one and might require solving an optimization problem.

[Osburg]

I have an idea how one could maybe do this if the dimension of the problem is low enough. One could determine the points as described in the sketch below and then sample from the set we computed. Inside each affine subspace one could just randomly draw coefficients for basis vectors, one would have to think about how to determine the probability for each subspace if they do not all have the same dimension... Do you think this could make sense ?

[DavidWalz]

From our discussion yesterday. While I think this is a feasible approach for small number of parameters, I'm afraid it would get too slow quickly. E.g for 10-choose-4 we have 210 subspace projections to iterate over and sample from. Additionally we would need to consider relative volumes for each of these subspaces to approach a uniform sampling.