Robust solutions to l1, l2, and l∞ uncertain linear approximation problems using convex optimization

We present minimax and stochastic formulations of some linear approximation problems with uncertain data in R² equipped with the Euclidean (l₂), absolute-sum (l₁) or Chebyshev (l_∞) 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, and the work of Ben-Tal and Nemirovski (1995) on robust conic convex optimization problems. The theory presented here is useful for desensitizing solutions to ill-conditioned problems, or for computing solutions that guarantee a certain performance in the presence of uncertainty in the data.