Robust least squares and applications

We consider least-squares problems where the coefficient matrices A,b are unknown-but-bounded. We minimize the worst-case residual error using (convex) second-order cone programming (SOCP), yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of the solution, and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g. Toeplitz) structure, the same problem can be solved in polynomial-time using semidefinite programming (SDP). We also consider the case when A,b are rational functions of an unknown-but-bounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worst-case residual. We provide numerical examples, including one from robust identification and one from robust interpolation.