# Robust Counterpart of an uncertain LP

Consider the following robust optimization problem:

min c'x

s.t.: $Ax\geq b \;\;\forall (A,b)\in \mathcal{U}$.

Why can the robust counterpart of the problem be written in this form? $min_x{\{ max_{(A,b)} Ax\geq b\}}$?

How does it work principally? Does it mean that I have to, for a fixed $x$ value, maximize the constraints $Ax\geq b$ for all realization of the uncertain data, and then minimize the value $x$?

Please make this point clear to me with a small example.

• Where do you see it written this way? This doesn't really make a lot of sense. For one thing, $Ax$ is a vector, and the $\max$ function as written doesn't really operate on vectors. Perhaps this is a convention adopted by some in the robust optimization community; if so, perhaps you can point us to a paper. – Michael Grant Apr 19 '13 at 1:11