Robust solutions of uncertain quadratic and conic-quadratic problems

We consider a conic-quadratic (and in particular a quadratically constrained) optimization problem with uncertain data, known only to reside in some uncertainty set U. The robust counterpart of such a problem leads usually to an NP-hard semidefinite problem; this is the case, for example, when U is given as the intersection of ellipsoids or as an n-dimensional box. For these cases we build a single, explicit semidefinite program, which approximates the NP-hard robust counterpart, and we derive an estimate on the quality of the approximation, which is essentially independent of the dimensions of the underlying conic-quadratic problem.