I would like to know how to determine all endpoints or intersections (vertices) of multidimenzional planes.
Let's say I have an object (for example multidimensional cube) given by the following sets of inequalities ($p$-times leq; $q$ times geq):
$$ \begin{bmatrix} a_{1,1} & \dots & a_{1,i} & \dots & a_{1,n} \\ a_{2,1} & \dots & a_{2,i} & \dots & a_{2,n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{p,1} & \dots & a_{p,i} & \dots & a_{p,n} \end{bmatrix} * \begin{bmatrix} x_{1} \\ \vdots \\ x_{i} \\ \vdots \\ x_{n} \end{bmatrix} \leq \begin{bmatrix} H_{1} \\ H_{2} \\ \vdots \\ H_{p} \end{bmatrix} $$
$$ \begin{bmatrix} b_{1,1} & \dots & b_{1,i} & \dots & b_{1,n} \\ b_{2,1} & \dots & b_{2,i} & \dots & b_{2,n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ b_{q,1} & \dots & b_{q,i} & \dots & b_{q,n} \end{bmatrix} * \begin{bmatrix} x_{1} \\ \vdots \\ x_{i} \\ \vdots \\ x_{n} \end{bmatrix} \geq \begin{bmatrix} L_{1} \\ L_{2} \\ \vdots \\ L_{q} \end{bmatrix} $$
Where $\boldsymbol{x}$ are variables, $\boldsymbol{a} $, $\boldsymbol{b} $, $\boldsymbol{L} $, and $\boldsymbol{H}$ are parameters: $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{L}, \boldsymbol{H} \in \mathbb{R}$.
In other words I have some angular object, for example as provided in this illustration, and would like to find all the endpoints. I am more interested in those violet ones, because considering additional constraints for nonnegativity, the object would be compact.
What I thought of yet is to use some modification of simplex algorithm. However, I am not sure how to modify it such that it identifies all the endpoints. Also, I am inquiring if there is something a little bit better and easier to implement.
Thank you very much.