### I am trying to answer the following question:

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

### This is what I tried:

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

$ V cap C = varnothing $ If and only if $ exist y in mathbb {R} ^ n $ such that

$$

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

$$

$$

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

$$

Written more compactly, $ V cap C = varnothing $ If and only if $ exist y in mathbb {R} ^ n $ such that

$$

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

$$

$$

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

$$

where $ 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 $ as its rows (i) can be treated "sufficiently" considering only $ z = e_1, …, e_k, e_1 + … + e_k $ since they cover positively $ mathbb {R} ^ k $. About (ii), to replace the infinitely many $ x $ finely many $ x $, **I would need the extreme rays of $ C $I don't know how to find**. But, even if I find these extreme rays, the problem will only become "similar" to the linear feasibility problem. **"With some $ <$ conditions ", which I still don't know how to treat completely**.

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

$$

Wx = 0

$$

$$

Ux <0

$$

$$

s.t. x in mathbb {R} ^ n

$$

It has no solution. Again, **I don't know how to deal with "$ <$"in $ Ux <0 $.**

### Sorry for my long paragraph that describes 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 analytical formulas to find their extreme rays?

(2) In the restrictions of a "linear program", if there are any restrictions with "$ <$" instead of "$ leq $"How do I deal with them?

Any ideas, reference books or documents are welcome and appreciated. Thank you.