Strictly positive linear combination of vectors

Question

I have a matrix S of size $m \times n$.

How to find a linar combination of the $n$ column vectors of the form:

$x = col(1) + \sum_{i=1}^{i=n}{\lambda_i col(i)}$ such that all entries of $x$ are strictly possitive

Thanks in advance

Answer 1

One way of doing it could be:

You can formulate this problem into finding an interior point of a polytope. x having strictly positive elements translates to the condition: Sλ+col(1)>0 or −Sλ−col(1)<0 (<,> is understood element-wise), which is describing the interior of the polytope P described by −Sλ−col(1)≤0. Now, there are many ways of finding an interior point of P. (Notice, there must not necessarily exist one). You could use an optimization solver like CVX to find a feasible point of the equations describing P for example.