I am working on optimization with linear inequality constraints. Suppose the constraints are given in matrix-vector form $A\mathbf{x} \geqslant \mathbf{b}$ and provide a non-empty feasible domain (convex polytope).
If I know that a certain boundary point of that domain is a result of intersection of a few hyperplanes $A_i\mathbf{x} = b_i$, where $A_i$ is the $i$-th row of matrix $A$, (those are active constraints) is there an algorithm that gives me a cone of feasible directions from that point? In other words, I am interested in finding a set of directions $\mathbf{d}$, such that, if going along those directions from that boundary point, I still remain in the feasible domain. To be specific, I need an algorithm that gives me a random vector from that cone based on the input $A_i\mathbf{x} = b_i$ for all $i$ relevant to the given boundary point.
For simplicity, we can assume that vectors $A_i^T$ (that are normal to their respective hyperplanes) point inside the feasible domain.