  The radius of robust feasibility provides a numerical value for the largest
possible uncertainty set that guarantees robust feasibility of an uncertain
linear conic program. This determines when the robust feasible set is
non-empty. Otherwise the robust counterpart of an uncertain program is not
well-defined as a robust optimization problem. In this paper, we address a key
fundamental question of robust optimization: How to compute the radius of
robust feasibility of uncertain linear conic programs, including linear
programs? We first provide computable lower and upper bounds for the radius of
robust feasibility for general uncertain linear conic programs under the
commonly used ball uncertainty set. We then provide important classes of linear
conic programs where the bounds are calculated by finding the optimal values of
related semidefinite linear programs (SDPs), among them uncertain SDPs,
uncertain second-order cone programs and uncertain support vector machine
problems. In the case of an uncertain linear program, the exact formula allows
us to calculate the radius by finding the optimal value of an associated
second-order cone program.
