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If I'm not mistaken this should be equivalent to the truncated moment problem for (not-necessarily homogeneous) bivariate quartics. Section 3 of "Positivity of Riesz Functionals and Solutions of Quad …

I'm assuming by interior you mean relative interior. If not just check first that both generating sets span the full space -- if not, one cone has empty topological interior and so the topological in …

As Yoav Kallus suggests you can also view this problem as finding a hyperplane which separates the $r_i$ from the $t_j$. For this you can use any algorithm for computing a linear classifier, such as …

This problem is clearly in NP (guess which polytopes) and becomes NP-complete if we replace $2/3$ with $1/2$ and make it a decision problem, dropping the promise that such a $p$ exists. In particular …

For the reasons discussed in the comments to Suvrit's answer, I will assume your friend would like to minimize $\det(\Lambda^{-1})$. The problem in question is a convex optimization problem: \begin{ …

Of course it's hard to say without any information, but linear matrix inequalities may be what you're looking for. They are to semidefinite programs what linear inequalities are to linear programs. …

The hit-and-run algorithm and variants are popular choices. These are Monte Carlo methods but should be much better than rejection sampling. Unfortunately I don't know of a canonical reference.

Edited in response to Alex Monras's correction in the comments: The cone $\mathcal{K}^*$ is always SDR: this is just the conic / homogeneous version of Theorem 5.57 in the new book "Semidefinite Opti …