Maximize the variance of sum of Bernoulli random variables

Question

Let $Z_1, \cdots, Z_n \in \{0,1\}$ be $n$ Bernoulli random variables, with known joint probability distribution. I am considering to solve the following problem in general: $$ \mathrm{maximize}_{x_1,\cdots,x_n \in \{0,1\}} {\mathbb{V}\mathrm{ar}}\Big[ \sum_{i=1}^{n}{x_i Z_i} \Big] $$ i.e., select a subset of random variables such that their summation has the largest variance. I feel the intuition is to select a subset of variables that are positively correlated. I wonder if this is a known problem and I am really looking forward to getting some thoughts here.

Answer 1

If you know the joint distribution, then you know the covariance matrix $\Sigma$ (whose $i,j$ entry is $\text{Cov}(Z_i, Z_j)$). Then the variance of $x^\top Z = \sum_{i=1}^n x_i Z_i$ is $$x^\top \Sigma x.$$ If the maximization problem were constrained to be in the unit ball $\|x\|=1$, then the maximizing $x$ would be the eigenvector corresponding to the largest eigenvalue of $\Sigma$.

However, your optimization problem is combinatorial; I believe it can be cast as a binary optimization problem. The linked PDF has some algorithms to solve them.