### I'm trying to answer the following question:

Let $ V = text {span} (v_1, …, v_k) subseteq mathbb {R} ^ n $, or $ v_1, …, v_k $ can be assumed orthonormal. Let $ C = {x in mathbb {R} ^ n: x ^ Tu_i <0, i = 1, …, N } $ to be an open polyhedral cone, where $ u_1, …, u_N in mathbb {R} ^ n $ are given. Suppose we have $ v_1, …, v_k $ and $ u_1, …, u_N $ in hand, how do we decide if $ V cap C = $ garment or not?

### Here is what I tried:

**My first approach** is to use the separation theorem to try to turn the problem into a standard linear feasibility problem:

$ V cap C = $ garment if and only if $ exists y in mathbb {R} ^ n $ such as

$$

y ^ T (r_1v_1 + cdots + r_kv_k) geq 0, forall (r_1, …, r_k) in mathbb {R} ^ k text {and}

$$

$$

y ^ Tx <0, forall x: x ^ Tu_i <0, i = 1, …, N.

$$

Written in a more compact way, $ V cap C = $ garment if and only if $ exists y in mathbb {R} ^ n $ such as

$$

(i) y ^ TA_V z geq 0, forall z = (z_1, …, z_k) ^ T in mathbb {R} ^ k text {, and}

$$

$$

(ii) y ^ Tx <0, forall x in mathbb {R} ^ n: Ux <0, i = 1, …, N

$$

or $ A_V $ is the $ n times k $ matrix with $ v_1, …, v_k $ like its columns and $ U $ is the $ N times n $ matrix with $ v_1, …, u_N $ like his rows. (i) can be "sufficiently" treated by considering only $ z = e_1, …, e_k, e_1 + … + e_k $ because they extend positively $ mathbb {R} ^ k $. About (ii), to replace the infinitely many $ x $ by finite number $ x $, **I would need the extreme rays of $ C $, that I do not know how to find**. But even if I find these extreme rays, the problem will only become something "similar" to a linear feasibility problem **"with some $ <$ "conditions," which I still do not know how to handle completely**.

**My second approach** is simply to consider $ V ^ perp $, the orthogonal complement of $ V $. We can use the Gram-Schmidt process to generate an orthonormal basis for $ V ^ perp $say $ w_1, …, w_ {n-k} $. Let $ W $ Be the $ (n-k) times n $ matrix with $ w_1, …, w_ {n-k} $ like his rows and $ U $ Be the $ N times n $ matrix with $ u_1, …, u_N $ like his rows. then $ V cap C = $ garment if and only if the system

$$

Wx = 0

$$

$$

Ux <0

$$

$$

x in mathbb {R} ^ n

$$

There is no solution. Again, **I do not know how to deal with "$ <$" in $ Ux <$ 0.**

### Sorry for my long paragraph describing my attempts. Here are my questions:

(1) For a polyhedral cone described by $ C = {x in mathbb {R} ^ n: x ^ Tu_i <0, i = 1, …, N } $ with $ u_i $ known, $ i = 1, …, N $Are there simple algorithms or analytic formulas for finding its extreme rays?

(2) In the constraints of a "linear program", there are constraints with "$ <$" instead of "$ leq $", how do I treat them?

Any idea, reference book, or papers are welcome and appreciated. Thank you.