Robust Solutions to l 1 , l 2 , and l 1 Uncertain LinearApproximation Problems using Convex Optimization 1

We present minimax and stochastic formulations of some linear approximation problems with uncertain data in R n equipped with the Euclidean (l2), Absolute-sum (l1) or Chebyshev (l1) norms. We then show that these problems can be solved using convex optimization. Our results parallel and extend the work of El-Ghaoui and Lebret on robust least squares 3], and the work of Ben-Tal and Nemirovski on robust conic convex optimization problems 1]. The theory presented here is useful for desensitizing solutions to ill-contitioned problems, or for computing solutions that guarantee a certain performance in the presence of uncertainty in the data.