# Positive linear combination of vectors to produce a positive vector

Given a list of vectors, I want a linear combination with positive coefficients that produces a final vector with only positive values (EDIT: this final vector is unknown; any positive vector is viable). If it's known that such a solution exists, any suggestions on an algorithm to find such a combination?

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This can be expressed as a linear program. – copper.hat Jul 3 '13 at 18:59
Just to make sure I understood, are you trying to solve $\sum_k \alpha_k v_k = b$, where $\alpha_k \ge 0$? Or are you trying to see if the convex cone generated by the vectors intersects the interior of the positive quadrant? – copper.hat Jul 3 '13 at 19:10
Where b is unknown, yes. Any positive a that produces a positive b is acceptable. (editing question for clarity) – akroy Jul 3 '13 at 20:22
Solve $\min_{\alpha,x} \{ \alpha | -e_k^T A x -\alpha \le 0, x \ge 0 \}$. If the minimum is $<0$ then a solution exists, otherwise it doesn't. – copper.hat Jul 3 '13 at 20:39
It's known that a solution exists, I'm curious how to find it. (out of curiosity, what does that expression actually represent? What is the e_k there?) – akroy Jul 3 '13 at 21:10

Put all vectors in the list as columns of the matrix $A$ and the result vector as a the column-vector $\vec{b}$. You seek such $\vec{x}$ that $A \vec{x} = \vec{b}$.
The OP needs the constraint $x \ge 0$ as well. – copper.hat Jul 3 '13 at 19:09