In this paper we consider semidefinite programs (SDPs) whose data depend on
some unknown but bounded perturbation parameters. We seek "robust" solutio
ns to such programs, that is, solutions which minimize the (worst-case) obj
ective while satisfying the constraints for every possible value of paramet
ers within the given bounds. Assuming the data matrices are rational functi
ons of the perturbation parameters, we show how to formulate sufficient con
ditions for a robust solution to exist as SDPs. When the perturbation is "f
ull," our conditions are necessary and sufficient. In this case, we provide
sufficient conditions which guarantee that the robust solution is unique a
nd continuous (Holder-stable) with respect to the unperturbed problem's dat
a. The approach can thus be used to regularize ill-conditioned SDPs. We ill
ustrate our results with examples taken from linear programming, maximum no
rm minimization, polynomial interpolation, and integer programming.