I am wondering what is known about optimization problems of the following type.

Our control x is a **unit vector** in $\mathbb{R}^n$. We are given a finite number of linear inequalities
$$Az≥b,$$
and we would like to select $x$ so that when substituted in place of $z$ the largest number of such inequalities is satisfied.

I can think of a brute force way of solving the problem, which is to consider every possible subset of the inequalities, and solve the optimization problem of finding the nearest point to the origin. Is there something better?

Any ideas or references to literature would be appreciated. I would also be interested in dual problems, and would like to know if this sort of problem has an official name or can be converted to a standard one.