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Given a matrix $A$ I want to find a vector $\vec{x}$ such that every element of $A\vec{x}$ is strictly positive. Also, the columns of $A$ do not span the full space, so if I were to just naively pick some $\vec{y}$ with all positive entries, I could not in general find a solution for $A\vec{x} = \vec{y}$. Is there a method guaranteed to find a feasible point if one exists? Bonus points if it's a matrix-free method, so I only have to evaluate matrix-vector products, no pseudoinverses or factorizations. I suspect that this can be set up as a convex program, but I'm not sure how.

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    Equivalently, you want a vector $x$ that has positive dot product with every *row* of the matrix. So it's a question of whether the intersection of open halfspaces determined by the row vectors is nonempty.2012-07-26

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