Characterizing robust local error bounds for linear inequality systems under data uncertainty

We present necessary and sufficient conditions for the existence of robust local error bounds for linear inequality system in the face of data uncertainty where the uncertain data belong to a prescribed compact uncertainty set. The robust local error bound for an uncertain system is defined in terms of the existence of local error bound for its robust counterpart, where the uncertain linear inequality is enforced for every possible value of the data in the uncertainty set. Some of these conditions are expressed using the projection of the origin and others are given by way of normal cones at the boundary points of the solution set. In the case of commonly used interval data uncertainty, we show that a qualification condition completely characterizes the existence of robust local error bounds.