Efficiently split a point cloud into two parts by a hyperplane to maximize the total sum of values associated with one part
The special case of your problem in which the points are in $\{-1,1\}^n$ and the values are $\pm 1$ is known as Maximum Agreement for Halfspaces. When talking about approximation, we want a positive objective function, and in this case it is convenient to ask for the number of correctly classified points. A point is correctly classified if either its value is $+1$ and it belongs to $S$, or its value is $-1$ and it belongs to $S^c$.

A trivial $1/2$-approximation algorithm just picks any halfspace and chooses either the halfspace or its complement. One of these satisfies at least half the constraints. Guruswami and Raghavendra showed that no better polynomial time approximation algorithm exists unless $\mathsf{P}=\mathsf{NP}$.

In practice you might be able to use some heuristics, but these of course don't provide any guarantees in the worst case.