Given an ordered input set of boolean values $S$ of length $|S|$,
I want to minimize a function $f(S)$
while maximizing the number of $True \in S$
constrained by $|True \in S|$ $\ge$ $threshold$
I understand this is NP-hard to manage with the $2^{|S|}$ possible inputs, but know some of them will be nonsensical.
Is there a way to frame this problem as a bayesian optimization problem for generating a set of booleans which minimize $f$ in such a way that I can avoid most of this parameter space?
Bayesian optimization over binary vectors is still a problem of major research. One potential way was proposed in https://arxiv.org/pdf/1806.08838.pdf. The key idea here is to represent the surrogate model as a second order model $\sum_i \alpha_j x_j + \sum_i \sum_j \alpha_{ij} x_i x_j$ which allows the acquisition function optimization to be approximately solved by a semidefinite relaxation.
Regarding your constraints $|True \in S| \geq threshold$, local search allows you to search over vectors which satisfies the constraints but might be sub-optimal.
The source code of above paper is available here https://github.com/baptistar/BOCS.
