We have $n$ variables $a\_0,a\_1,\ldots,a\_n$ such that $a\_i\geq a\_{i+1}$. 
There are $k$ sets of linear inequality constraints on the $a\_i$. 
I need to check that any choice of $a\_i$ satisfies at least one of the k sets of constraints.


If I see it correctly, essentially I would need to check that the union of the feasible sets of the linear inequalities covers the subset defined by the $a\_i$ of interest. 


What would be the simplest way to do this?


$k=5$, $n=5$ and each set of constraints has about 5~10 (maybe redundant) constraints. 



Union of linear inequalities cover whole space?
I found the solution for the problem. 
In summary, one has to check that the union of the polyhedra is convex. 
If their union is convex, the union is equal to their envelope. 
These problems can be solved using linear programming. 


Convexity recognition of the union of polyhedra
<http://www.sciencedirect.com/science/article/pii/S0925772101000049>


