I have a set of vectors $\{V_i\}$ in $n$-dimensional space. There is a number corresponded to each vector $\alpha_i = f(V_i)$ ($\alpha_i$ can be negative). I want to find a hyperplane which would maximize the difference between sums of $\alpha_i$ on the different sides of the space, divided by plane.
What is the best way to do this?
Here is a model for your problem. I assumed that the points that lie in the hyperplane do not count towards the sum, otherwise a slightly modification of the solution below will work.
Let $H=\{x\in \mathbb{R}^n: \bar{c}^Tx=\bar{\alpha}\}$ be the desired hyperplane. Then $(\bar{c},\bar{\alpha}) \in \mathbb{R}^{n+1}$ is an optimal solution of the problem
$$\max_{(c,\alpha) \in \mathbb{R}^{n+1}} \left|\sum_{i=1}^k \text{sgn}(c^Tv_i -\alpha) \alpha_i\right|,$$ where
$$sgn(t)= 1, \;t>0,\; sgn(t)=-1,\;t<0,\textrm{ and }sgn(0)=0.$$