Robust Solutions for Systems of Uncertain Linear Equations

Our contribution is twofold. Firstly, for a system of uncertain linear equations where the uncertainties are column-wise and reside in general convex sets, we show that the intersection of the set of possible solutions and any orthant is convex. We derive a convex representation of this intersection to calculate the ranges of the coordinates. Secondly, we propose two new methods for obtaining robust solutions of systems of uncertain linear equations. The first method calculates the center of the maximum inscribed ellipsoid of the set of possible solutions. The second method minimizes the expected violations with respect to the worst-case distribution. We compare these two new methods both theoretically and numerically with an existing method. The existing method minimizes the worst-case violation. Applications to the input-output model, Colley's Matrix Rankings and Article Influence Scores demonstrate the advantages of the two new methods.