Suppose I have a number of vectors in $\mathbb{R}^n.$ The first question is: what is the most efficient algorithm to compute their "conic hull" (the minimal convex cone which contains them)? The next question is: suppose I have a number of vectors $v_1, \dotsc, v_n,$ as before, and a convex cone $C.$ I want to find the conic hull of $\{v_1, \dotsc, v_n\} \cup C.$ In case it matters, in my application $C$ is the semidefinite cone. By "compute the conic hull", I mean: I want to find the subset of the $v_i$ on the boundary of the hull.