I have a number of questions of varying difficulties related to satisfying largish (as large as possible) systems of linear inequalities. I gather these aren't easy, so I'd be happy to get numerical algorithms or helpful references. The questions are motivated by research into the mechanics of large random systems with one-way constraints (the good news is that I'm a physicist, so I don't need to be too rigorous and if I get it wrong nobody's going to die). In order of increasing difficulty:
How can I tell whether a linear subspace nontrivially intersects with one or more hyperoctants? That is, for a given largish real matrix $$\mathbf{M}$$ how can I tell whether there is some nonzero (one could specify unit) vector $$\mathbf{c}$$ such that all or some of the elements of $$\mathbf{M}\mathbf{c}$$ are positive? Does it matter if I need them to be strictly positive or only nonnegative?
What if instead I have some new matrix $$\mathbf{M}'$$ and I instead want to know whether there is a solution to the equality $$\mathbf{M}' \mathbf{c}=0$$ subject to the requirement that all or some of the elements of $$\mathbf{c}$$ are positive? Again, does it matter whether the positivity is strict or weak?
Can we make any estimates of what the answers are if the matrix columns point in random (iid) directions? For example, is there something like a result that says that as the number of inequalities becomes large you almost certainly need twice as many elements of $$\mathbf{c}$$ as you have inequality constraints?
What about if we add a source term and/or some equalities, so that we need the elements of $$\mathbf{M}' \mathbf{c}$$ to exceed particular nonzero values, or to equal a particular nonzero values?