How to compute the intersection of a subspace and a cone?

Question

Let $A\in \mathbb R^{m\times n}$ and let

$$S= \{y\in \mathbb R^n: y^T= \lambda^TA,\ \lambda\in \mathbb R^m\}$$ be the row space spanned by the rows of $A$. Let $$C= \{x\in \mathbb R^n: Ax\leq 0\}$$ be a cone.

How can I compute the intersection $S\cap C$?

Thanks.