For a matrix A such that $Ax \ge 0$ has no non zero solution, find the closest matrix A' to A such that $A'x \ge 0$ has non zero solutions.

Question

We know that for a matrix A where $Ax = 0$ has no non zero solution, the smallest eigenvalue of A represents the distance from A to the closest matrix A' such that $A'x = 0$ has non zero solutions. Is there an analogous measure for the problem described in the title?