Robust solutions of perturbed linear equations

A perturbed system of linear equalities
\langle a_{i},x \rangle = b_{i}, i = 1,2,...,n;a_{i} \inA_{i};b_{i},\inB_{i};x\inX
(the sets A
i
and the intervals B
i
prescribed a priori) is said to be robust if a solution vector
x_{0}\inX
can be found resulting in
\langle a_{i},x_{0}\rangle \in B_{i}
for all
a_{i} \inA_{i}
and all
i = 1, 2,...,n
. A numerical "test for robustness" is developed. This test is seen to involve 2n parameters at most-even when the solution set
X
is an infinite-dimensional vector space.